$$\require{cancel}$$

# 14.3: Temperature Change and Heat Capacity

One of the major effects of heat transfer is temperature change: heating increases the temperature while cooling decreases it. We assume that there is no phase change and that no work is done on or by the system. Experiments show that the transferred heat depends on three factors—the change in temperature, the mass of the system, and the substance and phase of the substance.

Figure $$\PageIndex{1}$$: The heat $$Q$$ transferred to cause a temperature change depends on the magnitude of the temperature change, the mass of the system, and the substance and phase involved. (a) The amount of heat transferred is directly proportional to the temperature change. To double the temperature change of a mass $$m$$, you need to add twice the heat. (b) The amount of heat transferred is also directly proportional to the mass. To cause an equivalent temperature change in a doubled mass, you need to add twice the heat. (c) The amount of heat transferred depends on the substance and its phase. If it takes an amount $$Q$$ of heat to cause a temperature change $$\Delta T$$ in a given mass of copper, it will take 10.8 times that amount of heat to cause the equivalent temperature change in the same mass of water assuming no phase change in either substance.

The dependence on temperature change and mass are easily understood. Owing to the fact that the (average) kinetic energy of an atom or molecule is proportional to the absolute temperature, the internal energy of a system is proportional to the absolute temperature and the number of atoms or molecules. Owing to the fact that the transferred heat is equal to the change in the internal energy, the heat is proportional to the mass of the substance and the temperature change. The transferred heat also depends on the substance so that, for example, the heat necessary to raise the temperature is less for alcohol than for water. For the same substance, the transferred heat also depends on the phase (gas, liquid, or solid).

Heat Transfer and Temperature Change

The quantitative relationship between heat transfer and temperature change contains all three factors: $Q = mc\Delta T,$ where $$Q$$ is the symbol for heat transfer, $$m$$ is the mass of the substance, and $$\Delta T$$ is the change in temperature. The symbol $$c$$ stands for specific heat and depends on the material and phase. The specific heat is the amount of heat necessary to change the temperature of 1.00 kg of mass by $$1.00^oC$$. The specific heat $$c$$ is a property of the substance; its SI unit is $$J/(kg \cdot K)$$ or $$J/(kg \cdot ^oC)$$. Recall that the temperature change $$(\Delta T)$$ is the same in units of kelvin and degrees Celsius. If heat transfer is measured in kilocalories, then the unit of specific heat is $$kcal/(kg \cdot ^oC)$$.

Values of specific heat must generally be looked up in tables, because there is no simple way to calculate them. In general, the specific heat also depends on the temperature. Table $$\PageIndex{1}$$ lists representative values of specific heat for various substances. Except for gases, the temperature and volume dependence of the specific heat of most substances is weak. We see from this table that the specific heat of water is five times that of glass and ten times that of iron, which means that it takes five times as much heat to raise the temperature of water the same amount as for glass and ten times as much heat to raise the temperature of water as for iron. In fact, water has one of the largest specific heats of any material, which is important for sustaining life on Earth.

Example $$\PageIndex{1}$$: Calculating the Required Heat: Heating  Water in an Aluminum Pan

A 0.500 kg aluminum pan on a stove is used to heat 0.250 liters of water from $$20.0^oC$$ to $$80.0^oC$$.

(a) How much heat is required? What percentage of the heat is used to raise the temperature of (b) the pan and (c) the water?

Strategy

The pan and the water are always at the same temperature. When you put the pan on the stove, the temperature of the water and the pan is

increased by the same amount. We use the equation for the heat transfer for the given temperature change and mass of water and aluminum. The specific heat values for water and aluminum are given in Table $$\PageIndex{1}$$.

Solution

Because water is in thermal contact with the aluminum, the pan and the water are at the same temperature.

1. Calculate the temperature difference: $\Delta T = T_f - T_i = 60.0^oC.$
2. Calculate the mass of water. Because the density of water is $$1000 \space kg/m^3$$, one liter of water has a mass of 1 kg, and the mass of 0.250 liters of water is $$m_w = 0.250 \space kg$$.
3. Calculate the heat transferred to the water. Use the specific heat of water in Table $$\PageIndex{1}$$ $Q_w = m_wc_w\Delta T = (0.250 \space kg)(4186 \space J/kg ^oC)(60.0^oC) = 62.8 \space kJ.$
4. Calculate the heat transferred to the aluminum. Use the specific heat for aluminum in Table $$\PageIndex{1}\: $Q_{Al} = m_{Al}c_{Al}\Delta T = (0.500 \space kg)(900 \space J/kg^oC)(60.0^oC) = 27.0 kJ.$ 5. Compare the percentage of heat going into the pan versus that going into the water. First, find the total transferred heat: $Q_{Total} = Q_W + Q_{Al} = 62.8 \space kJ + 27.0 \space kJ = 89.8 \space kJ.$ 6. Thus, the amount of heat going into heating the pan is $\dfrac{62.8 \space kJ}{89.8 \space kJ} \times 100\% = 69.9\%$ Discussion In this example, the heat transferred to the container is a significant fraction of the total transferred heat. Although the mass of the pan is twice that of the water, the specific heat of water is over four times greater than that of aluminum. Therefore, it takes a bit more than twice the heat to achieve the given temperature change for the water as compared to the aluminum pan. Example \(\PageIndex{2}$$: Calculating the Temperature Increase from the Work Done on a Substance: Truck Brakes Overheat on Downhill Runs

Truck brakes used to control speed on a downhill run do work, converting gravitational potential energy into increased internal energy (higher temperature) of the brake material. This conversion prevents the gravitational potential energy from being converted into kinetic energy of the truck. The problem is that the mass of the truck is large compared with that of the brake material absorbing the energy, and the temperature increase may occur too fast for sufficient heat to transfer from the brakes to the environment.

Figure $$\PageIndex{2}$$: The smoking brakes on this truck are a visible evidence of the mechanical equivalent of heat.

Calculate the temperature increase of 100 kg of brake material with an average specific heat of $$800.0 \space J/kg \cdot ^oC$$ if the material retains 10% of the energy from a 10,000-kg truck descending 75.0 m (in vertical displacement) at a constant speed.

Strategy

If the brakes are not applied, gravitational potential energy is converted into kinetic energy. When brakes are applied, gravitational potential energy is converted into internal energy of the brake material. We first calculate the gravitational potential energy $$(Mgh)$$ that the entire truck loses in its descent and then find the temperature increase produced in the brake material alone.

Solution

1. Calculate the change in gravitational potential energy as the truck goes downhill $Mgh = (10,000 \space kg)(9.80 \space m/s^2)(75.0 \space m) = 7.35 \times 10^6 \space J.$
2. Calculate the temperature from the heat transferred using $$Q = Mgh$$ and $\Delta T = \dfrac{Q}{mc},$ where $$m$$ is the mass of the brake material. Insert the values $$m = 100 \space kg$$ and $$c = 800 \space J/kg \cdot ^oC$$ to find $\Delta T = \dfrac{(7.35v \times 10^6 \space J)}{(100 \space kg)(800 \space J/kg^oC)} = 92^oC.$

Discussion

This temperature is close to the boiling point of water. If the truck had been traveling for some time, then just before the descent, the brake temperature would likely be higher than the ambient temperature. The temperature increase in the descent would likely raise the temperature of the brake material above the boiling point of water, so this technique is not practical. However, the same idea underlies the recent hybrid technology of cars, where mechanical energy (gravitational potential energy) is converted by the brakes into electrical energy (battery).

Table $$\PageIndex{1}$$: Specific Heats1 of Various Substances
Substances Specific heat (c)
Solids $$J/kg\cdot^oC$$ $$kcal/kg\cdot^oC$$
Aluminum 900 0.215
Asbestos 800 0.19
Concrete, granite (average) 840 0.20
Copper 387 0.0924
Glass 840 0.20
Gold 129 0.0308
Human body (average at 37 °C) 3500 0.83
Ice (average, -50°C to 0°C) 2090 0.50
Iron, steel 452 0.108
Silver 235 0.0562
Wood 1700 0.4
Liquids
Benzene 1740 0.415
Ethanol 2450 0.586
Glycerin 2410 0.576
Mercury 139 0.0333
Water (15.0 °C) 4186 1.000
Gases 3
Air (dry) 721 (1015) 0.172 (0.242)
Ammonia 1670 (2190) 0.399 (0.523)
Carbon dioxide 638 (833) 0.152 (0.199)
Nitrogen 739 (1040) 0.177 (0.248)
Oxygen 651 (913) 0.156 (0.218)
Steam (100°C) 1520 (2020) 0.363 (0.482)

Note that Example $$\PageIndex{2}$$ is an illustration of the mechanical equivalent of heat. Alternatively, the temperature increase could be produced by a blow torch instead of mechanically.

Example $$\PageIndex{3}$$: Calculating the Final Temperature When Heat Is Transferred Between Two Bodies: Pouring Cold Water in a Hot Pan

Suppose you pour 0.250 kg of $$20.0^oC$$ water (about a cup) into a 0.500-kg aluminum pan off the stove with a temperature of $$150^oC$$. Assume that the pan is placed on an insulated pad and that a negligible amount of water boils off. What is the temperature when the water and pan reach thermal equilibrium a short time later?

Strategy

The pan is placed on an insulated pad so that little heat transfer occurs with the surroundings. Originally the pan and water are not in thermal equilibrium: the pan is at a higher temperature than the water. Heat transfer then restores thermal equilibrium once the water and pan are in contact. Because heat transfer between the pan and water takes place rapidly, the mass of evaporated water is negligible and the magnitude of the heat lost by the pan is equal to the heat gained by the water. The exchange of heat stops once a thermal equilibrium between the pan and the water is achieved. The heat exchange can be written as $$|Q_{hot}| = Q_{cold}.$$

Solution

1. Use the equation for heat transfer $$Q = mc\Delta T$$  to express the heat lost by the aluminum pan in terms of the mass of the pan, the specific heat of aluminum, the initial temperature of the pan, and the final temperature: $Q_{hot} = m_{Al}c_{Al}(T_f - 150^oC).$

2. Express the heat gained by the water in terms of the mass of the water, the specific heat of water, the initial temperature of the water and the final temperature: $Q_{cold} = m_wc_w(T_f - 20.0^oC).$

3. Note that $$Q_{hot}<0$$ and $$Q_{cold} >0$$ and that they must sum to zero because the heat lost by the hot pan must be the same as the heat gained by the cold water: $Q_{cold} + Q_{hot} = 0,$ $Q_{cold} = -Q_{hot},$ $m_wc_w(T_f - 20.0^oC) = -m_{Al}c_{Al}(T_f - 150.0^oC).$

4. This an equation for the unknown final temperature, $$T_f$$.

5. Bring all terms involving $$T_f$$ on the left hand side and all other terms on the right hand side.