11th Physics Volume

W = Area = V f V i P d V ( P f , V f ) ( P i , V i ) The shape of PV diagram depends on the nature of the thermodynamic process. Unit Heat and thermodynamics is called specific heat capacity at constant volume. If the volume is kept constant, then the supplied heat is used to increase only the internal energy. No work is done by the gas as shown in Figure .

. Figure . Specific heat capacity at constant volume Conductor Gas Q Insulator Pin It implies that to increase the temperature of the gas at constant volume requires less heat than increasing the temperature of the gas at constant pressure. In other words s p is always greater than s v .

Molar Specific heat capacities Sometimes it is useful to calculate the molar heat capacities C p and C v . The amount of heat required to raise the temperature of one mole of a substance by 1K or °C at constant volume is called molar specific heat capacity at constant volume (C v ). If pressure is kept constant, it is called molar specific heat capacity at constant pressure (C p ). the structure and molecular nature of the system.

Unlike solids and liquids, gases have two specific heats: specific heat capacity at constant pressure ( s p ) and specific heat capacity at constant volume ( s v ). . . Specific heat capacity Specific heat capacity at constant pressure ( s p ): The amount of heat energy required to raise the temperature of one kg of a substance by K or °C by keeping the pressure constant is called specific heat capacity of at constant pressure.

When the heat energy is supplied to the gas, it expands to keep the pressure constant as shown in Figure . Figure . Specific heat capacity at constant pressure Conductor Gas Q Insulator In this process a part of the heat energy is used for doing work (expansion) and the remaining part is used to increase the internal energy of the gas. Specific heat capacity at constant volume (s v ): The amount of heat energy required to raise the temperature of one kg of a substance by K or °C by keeping the volume constant Unit Heat and thermodynamics .

Q = µC p dT ( . ) If W is the workdone by the gas in this process, then W = PdV ( . ) But from the first law of thermodynamics, Q = dU + W ( . ) Substituting equations ( .

), ( . ) and ( . ) in ( . ), we get, µC p dT = µC v dT + PdV ( .

) For mole of ideal gas, the equation of state is given by PV = µRT ⇒ PdV+VdP = µRdT ( . ) Since the pressure is constant, dP= ∴ C p dT = C v dT +RdT ∴ C P = C v +R (or) C p - C v = R ( . ) This relation is called Meyer’s relation It implies that the molar specific heat capacity of an ideal gas at constant pressure is greater than molar specific heat capacity at constant volume. The relation shows that specific heat at constant pressure (s p ) is always greater that specific heat at constant volume (s v ).

THERMODYNAMIC PROCESSES . . Isothermal process It is a process in which the temperature remains constant but the pressure and volume of a thermodynamic system will change. The ideal gas equation is If Q is the heat supplied to mole of a gas at constant volume and if the temperature changes by an amount D T , we have Q = µ C v D T .

( . ) By applying the first law of thermodynamics for this constant volume process (W= , since dV= ), we have Q = D U - ( . ) By comparing the equations ( . ) and ( .

), D U = µ C v D T or C v = µ D D U If the limit D T goes to zero, we can write C v = µ dU dT ( . ) Since the temperature and internal energy are state variables, the above relation holds true for any process. . .

Meyer’s relation Consider µ mole of an ideal gas in a container with volume V, pressure P and temperature T. When the gas is heated at constant volume the temperature increases by dT. As no work is done by the gas, the heat that flows into the system will increase only the internal energy. Let the change in internal energy be dU.

If C v is the molar specific heat capacity at constant volume, from equation ( . ) dU = µC v dT ( . ) Suppose the gas is heated at constant pressure so that the temperature increases by dT. If‘Q’ is the heat supplied in this process and ‘dV’ the change in volume of the gas.

Unit Heat and thermodynamics For an isothermal process, the first law of thermodynamics can be written as follows, Q = W ( . ) From equation ( . ), we infer that the heat supplied to a gas is used to do only external work. It is a common misconception that when there is flow of heat to the system, the temperature will increase.

For isothermal process this is not true. The isothermal compression takes place when the piston of the cylinder is pushed. This will increase the internal energy which will flow out of the system through thermal contact. This is shown in Figure .

. Figure . Isothermal expansion and isothermal compression Isothermal compression Q < , W < Isothermal expansion Q > , W > Thermal contact PV = µ RT Here, T is constant for this process So the equation of state for isothermal process is given by PV = constant ( . ) This implies that if the gas goes from one equilibrium state ( P , V ) to another equilibrium state ( P , V ) the following relation holds for this process P V = P V ( .

) Since PV = constant, P is inversely proportional to V ( P ∝ V ). This implies that PV graph is a hyperbola. The pressure- volume graph for constant temperature is also called isotherm. Figure .

shows the PV diagram for quasi-static isothermal expansion and quasi-static isothermal compression. We know that for an ideal gas the internal energy is a function of temperature only. For an isothermal process since temperature is constant, the internal energy is also constant. This implies that dU or D U = .

Figure . (a) Quasi-static isothermal expansion (b) Quasi-static isothermal compression A ( P i , V i ) B ( P f , V f ) Isothermal Expansion (a) T = const A ( P f , V f ) B ( P i , V i ) Isothermal Compression (b) T = const P f P i V f V i P i P f V i V f Unit Heat and thermodynamics Since we have an isothermal expansion, i > , so ln V i   > . As a result the work done by the gas during an isothermal expansion is positive. The above result in equation ( .

) is true for isothermal compression also. But in an isothermal compression V i < . So ln V i   < . As a result the work done on the gas in an isothermal compression is negative.

In the PV diagram the work done during the isothermal expansion is equal to the area under the graph as shown in Figure . Figure . Work done in an isothermal process. Shaded area = work done during isothermal expansion A ( P i , V i ) B ( P f , V f ) Isothermal Expansion (a) T = const P i P f V i V f Shaded area = work done during isothermal compression A ( P f , V f ) B ( P i , V i ) Isothermal Compression (b) T = const P f P i V f V i Similarly for an isothermal compression, the area under the PV graph is equal to the Examples: (i) When water is heated, at the boiling point, even when heat flows to water, the temperature will not increase unless the water completely evaporates.

Similarly, at the freezing point, when the ice melts to water, the temperature of ice will not increase even when heat is supplied to ice. (ii) All biological processes occur at constant body temperature ( °C). Work done in an isothermal process: Consider an ideal gas which is allowed to expand quasi-statically at constant temperature from initial state ( P i , V i ) to the final state ( P f , V f ). We can calculate the work done by the gas during this process.

From equation ( . ) the work done by the gas, W = i