Classwork Series and Exercises {Physics – SS2}: Heat Capacity

Physics SS 2 Week 2

Topic: Heat Capacity

Introduction

The heat capacity measures the amount of heat necessary to raise the temperature of an object or system by one degree Celsius.

Heat capacity is defined as the ratio of the heat energy added to an object to its change in temperature.

Heat capacity is the measurable physical quantity that characterizes the amount of heat required to change a substance’s temperature by a given amount. It is measured in joules per Kelvin and given by C = Q/∆T

The heat capacity is an extensive property, scaling with the size of the system.

The heat capacity of most systems is not constant (though it can often be treated as such). It depends on the temperature, pressure, and volume of the system under consideration.

Enthalpy the total amount of energy in a system, including both the internal energy and the energy needed to displace its environment

Heat capacity (usually denoted by a capital C, often with subscripts), or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance’s temperature by a given amount. In SI units, heat capacity is expressed in units of joules per kelvin (J/K).

An object’s heat capacity (symbol C) is defined as the ratio of the amount of heat energy transferred to an object to the resulting increase in temperature of the object

Heat capacity is an extensive property, so it scales with the size of the system. A sample containing twice the amount of substance as another sample requires the transfer of twice as much heat (Q) to achieve the same change in temperature (ΔT). For example, if it takes 1,000 J to heat a block of iron, it would take 2,000 J to heat a second block of iron with twice the mass as the first.

The Measurement of Heat Capacity

The heat capacity of most systems is not a constant. Rather, it depends on the state variables of the thermodynamic system under study. In particular, it is dependent on temperature itself, as well as on the pressure and the volume of the system, and the ways in which pressures and volumes have been allowed to change while the system has passed from one temperature to another. The reason for this is that pressure-volume work done to the system raises its temperature by a mechanism other than heating, while pressure-volume work done by the system absorbs heat without raising the system’s temperature. (The temperature dependence is why the definition of a calorie is formally the energy needed to heat 1 g of water from 14.5 to 15.5 °C instead of generally by 1 °C.)

Different measurements of heat capacity can therefore be performed, most commonly at constant pressure and constant volume. The values thus measured are usually subscripted (by p and V, respectively) to indicate the definition. Gases and liquids are typically also measured at constant volume. Measurements under constant pressure produce larger values than those at constant volume because the constant pressure values also include heat energy that is used to do work to expand the substance against the constant pressure as its temperature increases. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.

Thermodynamic Relations and Definition of Heat Capacity

The internal energy of a closed system changes either by adding heat to the system or by the system performing work. Recalling the first law of thermodynamics,

For work as a result of an increase of the system volume we may write,

dU = dQ – PdV

If the heat is added at constant volume, then the second term of this relation vanishes and one readily obtains

(¶U/¶T)_V = (¶Q/¶T)_V = C_V

This defines the heat capacity at constant volume, C_V. Another useful quantity is the heat capacity at constant pressure, C_P. With the enthalpy of the system given by

H = U + PV

our equation for dU changes to

dH = ¶Q + VdP

and therefore, at constant pressure, we have

(¶H/¶T)_P = (¶Q/¶T)_P = C_P

Specific Heat Capacity

Specific heat capacity of a substance is defined as the heat capacity of the substance per unit mass of the substance. Therefore, if energy Q is given to the substance having mass m, and it results in the change in the temperature of the substance by ΔT. Hence, Specific Heat of the substance is

c = Q/mΔT

Heat capacity of a substance is denoted by C. It is defined as the amount of energy which is required to increase the temperature of the substance by 1°C.

Specific heat capacity (c) of a substance is the heat required to produce unit temperature rise in unit mass of the substance.

Specific Heat

When energy is given to a substance and the substance is not performing any work, it results into increase in temperature of the substance. (If the temperature of the substance is not increased, when the heat is given to it, and also the substance is not performing any work, it could lead to change in the phase of the substance and this phenomenon is termed as Phase Transition). Amount of heat, required to raise the temperature of a substance by a certain amount, depends on the properties of the substance and this amount of heat varies from substance to substance.

For example, the amount of energy or heat required to increase the temperature of 1 kg of water by 1°C is 4.186 J, but the amount of heat required to increase the temperature of 1 kg of copper by 1°C is only 387 J. There are two methods to increase the temperature of the substance:

1. By transferring heat or energy to it.

2. By doing Work on it.

According to the definition, if heat Q given to the substance increases the temperature of the substance by ΔT, then

Q = CΔT

Specific Heat Formula

Specific heat is essentially a measure of how thermally insensitive a substance is to the addition of energy. If the specific heat capacity of a substance is more, then for a given mass m of a substance, for a particular ΔT temperature change, more energy Q needs to be transferred to the substance as compared to the second substance having less specific heat capacity. Therefore, more the specific heat capacity, more energy needs to be transferred to the substance if the other conditions (Q, m,ΔT ) are same.

Formula for specific heat can be written asc =Q/mΔT

Specific Heat Units: SI unit of Specific Heat is Joules per Kilogram Kelvin. (J/kg.K).

Specific Heat Table

Specific Heat Table for some of the substances is given below at 25 ⁰C and Standard Atmospheric pressure

Substance	Specific Heat(J/kg. °C)
Specific Heat of Beryllium	1830
Specific Heat of Cadmium	230
Specific Heat of Copper	387
Specific Heat of Germanium	322
Specific Heat of Gold	129
Specific Heat of Iron	448
Specific Heat of Lead	128
Specific Heat of Silicon	703
Specific Heat of Silver	234
Specific Heat of Brass	380
Specific Heat of Glass	837
Specific Heat of Ice(-5°C)	2090
Specific Heat of Marble	860
Specific Heat of Wood	1700
Specific Heat of Alcohol(ethyl)	2400
Specific Heat of Mercury	140
Specific Heat of Water(15°C)	4186
Specific Heat of Steam(100°C)	2010
Specific Heat of Aluminium	900
Specific Heat of Tin	540
Specific Heat of Steel	120
Specific Heat of Sand	830
Specific Heat of Ethanol (Alcohol, ethyl 32°F)	2.3 K

Methods of determining Specific Heat Capacity

There are several simple methods for measuring the specific heat capacities of both solids and liquids, such as the method of mixtures, but we will consider here only electrical methods. Since the specific heat capacity varies with temperature, we have seen it is important to record the mean temperature at which the measurement is made.

Electrical calorimeters

Figure 1(a) and 1(b) show possible arrangements for electrical calorimeters for a solid and a liquid specimen.

The material under investigation is heated by an electrical immersion heater and the input energy (Q) and the rise in temperature that this produces are measured. If the mass of the specimen (solid or liquid) is m and its specific heat capacity C, then Q = m C (q₁ – q₀) + q

where θ₀ and θ₁ are the initial and final temperatures of the specimen and q is the heat loss. Using the cooling correction, the value of q may be found. This simple method can be used for liquids or solids, although in the case of a liquid, allowance has to be made for the thermal capacity of the container, and the liquid should also be stirred to allow an even distribution of the heat energy throughout its volume. This is necessary since liquids are such poor thermal conductors

The continuous-flow calorimeter

This was first developed by Calendar and Barnes in 1902 for the measurement of the specific heat capacity of a liquid, and is shown in diagram below. Its main advantage is that the thermal capacity of the apparatus itself need not be known.

Liquid flows in from a constant-head apparatus at a constant rate past a thermometer (θ ₀). It then flows around the heater coil and out past a second thermometer where the outlet temperature (θ₁) may be measured. When steady-state conditions have been reached (a temperature difference between inlet and outlet points of 5⁰C is reasonable) the temperatures and the flow rate of the liquid (m) are measured. A vacuum jacket round the heater coil reduces heat losses.

The electrical energy supplied to the heater coil (E = V I t) may be found readily with a joulemeter or with an ammeter and voltmeter.

Two sets of measurements are carried out.

For a first experiment we have:

Electrical energy supplied (E₁) = V₁ I₁ t₁ = m₁ C (θ₁ – θ₀) + q

C is the specific heat capacity of the liquid and q the heat loss to the surroundings and to the apparatus.

The flow rate and rate of energy input are now altered to give a second set of results. However, if the inlet and outlet temperatures are the same as in the first experiment the heat loss will also be the same. Therefore:

Electrical energy supplied (E₂) = = V₂ I₂ t₂ = m₂ C (θ₁ – θ_O) + q

Eliminating the heat loss (q) gives

Specific heat capacity of the liquid (C) = [fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”no-repeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][E₂ – E₁]/ (m₂ – m₁)(q₁ – q_o)

Practical advice

A smaller amount of water could be heated in a polystyrene cup than in a calorimeter; this reduces the heating time needed and provides insulation. The heater must be covered by the water. The heat absorbed by the polystyrene is also small compared to that absorbed by the calorimeter. However take care that the heater does not touch the cup or it will melt. Thermometers can also overbalance the cup. Always stir liquids before taking a temperature.

It is better to choose an immersion heater that fits all the way into the solid material rather than having part of it in the air. The top of the block should also be lagged. Take the highest temperature reached by the block after the heater has been switched off.

Questions

1.    The heat capacity measures the amount of heat necessary to raise the temperature of an object or system by ……………. degree celsius.

A. One B. Hundred C. Ninety   D. One thousand

2. Heat Capacity is measured in

A. Kelvin per joules B. joules per Kelvin C. Joules   D. Kelvin

3. The specific heat value for Aluminium in J/kg. °C is

A. 800 B. 900 C. 980 D. 450

4. Formula for Specific Heat Capacity can be written a s

A. Q/mΔT B. Qm∆T C. Q∆T D. m∆T

5. The heat capacity of most systems is not a constant. Rather, it depends on the state variables of the thermodynamic system under study. In particular, it is dependent on

A. Temperature only B. Pressure and Temperature   C. Volume and Pressure D. Temperature, Volume and Pressure.

 

Answers

1. A 2. B 3. B 4. A 5. D

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