Physics A Level
Chapter 19: Thermal physics 19.6 Calculating energy changes
Physics A Level
Chapter 19: Thermal physics 19.6 Calculating energy changes
So far, we have considered the effects of heating a substance in qualitative terms, and we have given an explanation in terms of a kinetic model of matter. Now we will look at the amount of energy needed to change the temperature of something, and to produce a change of state.
Specific heat capacity
If we heat some material so that its temperature rises, the amount of energy we must supply depends on three things, the:
- mass m of the material we are heating
- temperature change $\Delta q$ we wish to achieve
- material itself.
Some materials are easier to heat than others. It takes more energy to raise the temperature of $1 kg$ of water by ${1^ \circ }C$ than to raise the temperature of $1 kg$ of alcohol by the same amount.
We can represent this in an equation. The amount of energy E that must be supplied is given by:
$E = mc\Delta \theta $
where c is the specific heat capacity of the material.
Rearranging this equation gives:
$c = \frac{E}{{m\Delta \theta }}$
The specific heat capacity of a material can be defined as a word equation as follows:
Alternatively, specific heat capacity can be defined in words as follows:
$specific\,heat\,capacity = \frac{{energy\,\sup plied}}{{mass\,temperature\,change}}$
The numerical value of the specific heat capacity of a substance is the energy required per unit mass of the substance to raise the temperature by $1 K$ (or ${1^ \circ }C$).
The word ‘specific’ here means ‘per unit mass’; that is, per kg. From this form of the equation, you should be able to see that the units of c are $J\,k{g^{ - 1\,}}\,{K^{ - 1}}$ (or $J\,k{g^{ - 1\,}}\,{0^ \circ }{C^{ - 1}}$). Table 19.3 shows some values of specific heat capacity measured at ${0^ \circ }C$.
Specific heat capacity is related to the gradient of the sloping sections of the graph shown earlier in Figure 19.4. The steeper the gradient, the faster the substance heats up and hence the lower its specific heat capacity must be. Worked example 1 shows how to calculate the specific heat capacity of a substance.
| Substance | $c\,/\,J\,k{g^{ - 1}}\,{K^{ - 1}}$ |
| aluminium | 880 |
| copper | 380 |
| lead | 126 |
| glass | 500–680 |
| ice | 2100 |
| water | 4180 |
| seawater | 3950 |
| ethanol | 2500 |
| mercury | 140 |
Questions
8) You will need to use data from Table 19.3 to answer these questions.
Calculate the energy that must be supplied to raise the temperature of $5.0 kg$ of water from ${20^ \circ }C$ to ${100^ \circ }C$.
9) Which requires more energy – heating a $2.0 kg$ block of lead by $30 K$ or heating a $4.0 kg$ block of copper by $5.0 K$?
10) A well-insulated $1.2 kg$ block of iron is heated using a $50 W$ heater for $4.0 min$. The temperature of the block rises from ${22^ \circ }C$ to ${45^ \circ }C$. Find the experimental value for the specific heat capacity of iron.
PRACTICAL ACTIVITY 19.1
Determining specific heat capacity c
How can we determine the specific heat capacity of a material? The principle is simple: supply a known amount of energy to a known mass of the material and measure the rise in its temperature. Figure 19.14 shows one practical way of doing this for a metal.
The metal is in the form of a cylindrical block of mass $1.00 kg$. An electrical heater is used to supply the energy. This type of heater is used because we can easily determine the amount of energy supplied – more easily than if we heated the metal with a Bunsen flame, for example. An ammeter and voltmeter are used to make the necessary measurements.
A thermometer or temperature sensor is used to monitor the block’s temperature as it is heated. The block must not be heated too quickly; we want to be sure that the energy has time to spread throughout the metal.
The block should be insulated by wrapping it in a suitable material – this is not shown in the illustration. It would be possible, in principle, to determine c by making just one measurement of temperature change, but it is better to record values of the temperature as it rises and plot a graph of temperature $\theta $ against time t. The method of calculating c is illustrated in Worked example 2.
Sources of error
This experiment can give reasonably good measurements of specific heat capacities. As noted earlier, it is desirable to have a relatively low rate of heating, so that energy spreads throughout the block. If the block is heated rapidly, different parts may be at different temperatures.
Thermal insulation of the material is also vital. Inevitably, some energy will escape to the surroundings.
This means that more energy must be supplied to the block for each degree rise in temperature and so the experimental value for the specific heat capacity will be too high. One way around this is to cool the block below room temperature before beginning to heat it. Then, as its temperature rises past room temperature, heat losses will be zero in principle, because there is no temperature difference between the block and its surroundings.
WORKED EXAMPLE
2) An experiment to determine the specific heat capacity c of a $1.00 kg$ aluminium block is carried out; the block is heated using an electrical heater. The current in the heater is $4.17 A$ and the p.d. across it is $12 V$. Measurements of the rising temperature of the block are represented by the graph shown in Figure 19.15. Determine a value for the specific heat capacity c of aluminium.
Step 1: Write down the equation that relates energy change to specific heat capacity:
$E = mc\Delta \theta $
Step 2: Divide both sides by a time interval $\Delta t$:
$E = mc\left( {\frac{{\Delta \theta }}{{\Delta t}}} \right)$
The quantity $\frac{E}{{\Delta t}}$ is the rate at which energy is supplied; that is, the power P of the heater.
The quantity $\frac{{\Delta \theta }}{{\Delta t}}$ is the rise of temperature of the block; that is, the gradient of the graph of $\theta $ against t.
Hence: $P = m \times c \times gradient$
Step 3: Calculate the power of the heater and the gradient of the graph.
$\begin{array}{l}
Power = p.d\, \times \,current\\
P = VI = 12 \times 4.17\, \approx \,50W
\end{array}$
$\begin{array}{l}
gradient = \frac{{\Delta \theta }}{{\Delta t}}\\
= \frac{{16.4}}{{400}}\\
= {0.041^ \circ }\,C{s^{ - 1}}
\end{array}$
Step 4: Substitute values, rearrange and solve.
$\begin{array}{l}
50 = 1.00 \times c \times 0.041\\
c = \frac{{50}}{{(1.00 \times 0.041)}}\\
c = 1220\,J\,k{g^{ - 1}}\,{K^{ - 1}}
\end{array}$
Questions
11) At higher temperatures than shown, the graph in Figure 19.15 deviates increasingly from a straight line.
Suggest an explanation for this.
12) In measurements of the specific heat capacity of a metal, energy losses to the surroundings are a source of error. Is this a systematic error or a random error? Justify your answer.
13) In an experiment to measure the specific heat capacity of water, a student uses an electrical heater to heat some water. His results are shown. Calculate a value for the heat capacity of water. Comment on any likely sources of error.
$mass\,of\,bea\ker \, = \,150g$
$mass\,of\,bea\ker \, + water\, = \,672g$
$current\,in\,the\,heater\, = \,3.9A$
$p.d.\,acrss\,the\,heater\, = 11.4V$
$initial\,temperature\,{18.5^ \circ }C$
$final\,temperature\, = 30.2\,{\min ^ \circ }C$
$time\,taken\, = 13.2\,\min $
14) A block of paraffin wax was heated gently, at a steady rate. Heating was continued after the wax had completely melted. The graph of Figure 19.16 shows how the material’s temperature varied during the experiment.
a: For each section of the graph (AB, BC and CD), describe the state of the material.
b: For each section, explain whether the material’s internal energy is increasing, decreasing or
remaining constant.
c: Consider the two sloping sections of the graph. State whether the material’s specific heat capacity is
greater when it is a solid or when it is a liquid. Justify your answer.
Specific latent heat
Energy must be supplied to melt or boil a substance. (In this case, there is no temperature rise to consider since the temperature stays constant during a change of state.) This energy is called latent heat.
The numerical value of the specific latent heat of a substance is the energy required per kilogram of the substance to change its state without any change in temperature.
When a substance melts, this quantity is called the specific latent heat of fusion; for boiling, it is the specific latent heat of vaporisation.
To calculate the amount of energy E required to melt or vaporise a mass m of a substance, we simply need to know its specific latent heat L:
$E = mL$
L is measured in $J\,k{g^{ - 1}}$. (Note that there is no ‘per ${0^ \circ }C$’ since there is no change in temperature.) For water, the values are:
- specific latent heat of fusion of water, $330\,kJ\,k{g^{ - 1}}$
- specific latent heat of vaporisation of water, $2.26MJ\,k{g^{ - 1}}$
You can see that L for boiling water to form steam is roughly seven times the value for melting ice to form water. As we saw previously in the topic on heating ice, this is because, when ice melts, only one or two bonds are broken for each molecule; when water boils, several bonds are broken per molecule. Worked example 3 shows how to calculate these amounts of energy.
Questions
15) The specific latent heat of fusion of water is $330\,kJ\,k{g^{ - 1}}$. Calculate the energy needed to change $2.0 g$ of ice into water at ${0^ \circ }C$. Suggest why the answer is much smaller than the amount of energy calculated in Worked example 3.
A sample of alcohol is heated with a $40 W$ heater until it boils. As it boils, the mass of the liquid decreases at a rate of $2.25 g$ per minute. Assuming that $80\% $ of the energy supplied by the heater is transferred to the alcohol, estimate the specific latent heat of vaporisation of the alcohol. Give your answer in $kJ\,k{g^{ - 1}}$.
PRACTICAL ACTIVITY 19.2
Determining specific latent heat L
The principle of determining the specific latent heat of a material is similar to determining the specific heat capacity (but remember that there is no change in temperature).
Figure 19.17 shows how to measure the specific latent heat of vaporisation of water. A beaker containing water is heated using an electrical heater. A wattmeter (or an ammeter and a voltmeter) determines the rate at which energy is supplied to the heater. The beaker is insulated to minimise energy loss, and it stands on a balance. A thermometer is included to ensure that the temperature of the water remains at ${100^ \circ }C$.
The water is heated at a steady rate and its mass recorded at equal intervals of time. Its mass decreases as it boils.
A graph of mass against time should be a straight line whose gradient is the rate of mass loss. The wattmeter shows the rate at which energy is supplied to the water via the heater. We thus have:
$specific\,latent\,heat = \frac{{rate\,of\,\sup ply\,of\,energy}}{{rate\,of\,loss\,mass}}$
A similar approach can be used to determine the specific latent heat of fusion of ice. In this case, the ice is heated electrically in a funnel; water runs out of the funnel and is collected in a beaker on a balance.
As with any experiment, we should consider sources of error in measuring L and their effects on the final result. When water is heated to produce steam, some energy may escape to the surroundings so that the measured energy is greater than that supplied to the water. This systematic error gives a value of L, which is greater than the true value. When ice is melted, energy from the surroundings will conduct into the ice, so that the measured value of L will be an underestimate.
EXAM-STYLE QUESTIONS
1) The first law of thermodynamics can be represented by the expression: $\Delta U = q + W$.
An ideal gas is compressed at constant temperature.
Which row shows whether $\Delta U$, q and W are negative, positive or zero during the change? [1]
| $\Delta U$ | q | W | |
| A | negative | negative | positive |
| B | positive | positive | negative |
| C | zero | negative | positive |
| D | zero | positive | negative |
2) What is the internal energy of an object? [1]
A: the amount of heat supplied to the object
B: the energy associated with the random movement of the atoms in the object
C: the energy due to the attraction between the atoms in the object
D: the potential and kinetic energy of the object.
3) Describe the changes to the kinetic energy, the potential energy and the total internal energy of the molecules of a block of ice as:
a: it melts at ${0^ \circ }C$ [3]
b: the temperature of the water rises from ${0^ \circ }C$ to room temperature. [3]
[Total: 6]
4) Explain, in terms of kinetic energy, why the temperature of a stone increases when it falls from a cliff and lands on the beach below. [3]
5) Explain why the barrel of a bicycle pump gets very hot as it is used to pump up a bicycle tyre. (Hint: the work done against friction is not large enough to explain the rise in temperature.) [3]
6) The so-called ‘zeroth law of thermodynamics’ states that if the temperature of body A is equal to the temperature of body B and the temperature of body B is the same as body C, then the temperature of body C equals the temperature of body A.
7) a: Explain, in terms of energy flow, why the concept of temperature would be meaningless if this law was not obeyed. [2]
The first law of thermodynamics can be represented by the expression: $\Delta U = q + W$.
State what is meant by all the symbols in this expression. [3]
b: Figure 19.18 shows a fixed mass of gas that undergoes a change from A to B and then to C.
i- During the change from A to B, $220 J$ of thermal energy (heat) is removed from the gas. Calculate the change in the internal energy of the gas. [2]
ii- During the change from B to C, the internal energy of the gas decreases by $300 J$. Using the first law of thermodynamics explain how this change can occur. [2]
[Total: 7]
When a thermocouple has one junction in melting ice and the other junction in boiling water it produces an e.m.f. of $63\mu V$.
a: What e.m.f. would be produced if the second junction was also placed in melting ice? [1]
b: When the second junction is placed in a cup of coffee, the e.m.f. produced is $49\mu V$. Calculate the temperature of the coffee. [2]
c: The second junction is now placed in a beaker of melting lead at ${327^ \circ }C$.
Calculate the e.m.f. that would be produced. [2]
State the assumption that you make. [1]
[Total: 6]
9) The resistance of a thermistor at ${0^ \circ }C$ is $2000\Omega $. At ${100^ \circ }C$ the resistance falls to $200\Omega $. When the thermistor is placed in water of constant temperature, its resistance is $620\Omega $.
a: Assuming that the resistance of the thermistor varies linearly with temperature, calculate the temperature of the water. [2]
b: The temperature of the water on the thermodynamic scale is $280 K$.
By reference to the particular features of the thermodynamic scale of temperature, comment on your answer to part a. [3]
[Total: 5]
10) a: A $500 W$ kettle contains $300 g$ of water at ${20^ \circ }C$. Calculate the minimum time it would take to raise the temperature of the water to boiling point. [5]
b: The kettle is allowed to boil for 2 minutes. Calculate the mass of water that remains in the kettle. State any assumptions that you make. [4]
(Specific heat capacity of water $ = 4.18 \times {10^3}\,J\,k{g^{ - 1}}\,\,\,{0^ \circ }{C^{ - 1}}$; specific latent heat of vaporisation of water $ = 2.26 \times {10^6}\,J\,k{g^{ - 1}}$.)
[Total: 9]
11) a: Define specific heat capacity of a substance. [2]
b: A mass of $20 g$ of ice at $ - {15^ \circ }C$ is taken from a freezer and placed in a beaker containing 200 g of water at ${26^ \circ }C$. Data for ice and water are given in Table 19.5.
| Specific heat capacity / $J\,k{g^{ - 1}}\,{K^{ - 1}}$ | Specific latent heat of fusion / $J\,k{g^{ - 1}}$ | |
| ice | $2.1 \times {10^3}$ | $3.3 \times {10^5}$ |
| water | $4.2 \times {10^3}$ |
i- Calculate the amount of thermal energy (heat) needed to convert all the ice at $ - {15^ \circ }C$ to water at ${0^ \circ }C$. [2]
ii- Calculate the final temperature T of the water in the beaker assuming that the beaker has negligible mass. [3]
[Total: 7]
12) a: Define specific latent heat and explain the difference between latent heat of fusion and latent heat of vaporisation. [3]
b: An electric heater generating power of $120 W$ is immersed in a beaker of liquid that is placed on a balance. When the liquid begins to boil it is noticed that the mass of the beaker and liquid decreases by $6.2 g$ every minute.
i- State how this shows that the liquid is boiling at a steady rate. [1]
ii- Calculate a value for the specific latent heat of vaporisation of the liquid. [2]
iii- State and explain whether the value determined in ii is likely to be larger or smaller than the accepted value. [2]
[Total: 8]
13) a: Explain why energy is needed for boiling even though the temperature of the liquid remains constant. [2]
This diagram shows an apparatus that can be used to measure the specific latent heat of vaporisation of nitrogen.
b: Suggest why the nitrogen is contained in a vacuum flask. [1]
c: The change in mass of the nitrogen is measured over a specific time interval with the heater switched off. The heater is switched on, transferring energy at $40 W$, and the change of mass is found once more.
The results are shown in the table.
| Initial reading on balance / g | Final reading on balance / g | Time / minutes | |
| heater off | 834.7 | 825.5 | 4 |
| heater on | 825.5 | 797.1 | 2 |
Calculate the specific latent heat of vaporisation of liquid nitrogen. [4]
[Total: 7]
14) a: i- Explain what is meant by internal energy. [2]
ii- Explain what is meant by the absolute zero of temperature. [2]
b: An electric hot water heater has a power rating of $9.0 kW$. The water is heated as it passes through the heater. Water flows through the heater at a speed of $1.2\,m\,{s^{ - 1}}$ through pipes that have a total cross-sectional area of $4.8 \times {10^{ - 5}}\,{m^2}$. The temperature of the water entering the heater is ${15^ \circ }C$.
i- Calculate the mass of water flowing through the heater each second. [2]
ii- Calculate the temperature at which the water leaves the heater. [3]
iii- State any assumptions you have made in your calculation. [1]
iv- It is possible to adjust the temperature of the water from the heater.
Suggest how the temperature of the water could be increased without increasing the power of the heater. [1]
(Density of water $ = 1000\,kg\,{m^{ - 3}}$; specific heat capacity of water =$ = 4200\,J\,k{g^{ - 1}}\,{\,^ \circ }{C^{ - 1}}$.)
[Total: 11]
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| understand internal energy, determined by the state of the system, as the sum of a random distribution of kinetic and potential energies of the molecules of a system | 19.3 | |||
| relate a rise in temperature of an object to an increase in its internal energy | 19.3 | |||
| recall and use $W = p\Delta V$ for the work done when the volume of a gas changes at constant pressure and understand the difference between work done by a gas and the work done on a gas | 19.3 | |||
| recall and use the first law of thermodynamics: $\Delta U = q + W$ | 19.3 | |||
| understand that (thermal) energy is transferred from a region of higher temperature to a region of lower temperature and that regions of equal temperature are in thermal equilibrium | 19.4 | |||
| understand the use of a physical property that varies with temperature to measure temperature and state examples of such properties | 19.5 | |||
| understand that thermodynamic temperature does not depend on the property of any particular substance and recall and use: $T/K = \theta /{\,^ \circ }C + 273/15$ |
19.4 | |||
| understand that the lowest possible temperature is zero kelvin, absolute zero |
19.4 | |||
| define and use specific heat capacity 19.6 | 19.6 | |||
| define and use specific latent heat and distinguish between specific latent heat of fusion and specific latent heat of vaporisation. | 19.6 |