Reversible adiabatic change temperature

For many purposes it is more useful to develop an expression relating the temperature to the pressure in a reversible, adiabatic change. Since an ideal gas is under consideration, it follows that P Vi = RTi and P%V2 = RTt if these equations are combined with (10.5) so as to eliminate Vi and Vt, it is found that... 

In an adiabatic expansion of a gas, mechanical work is done by the gas as its volume increases and the gas temperature falls. For an ideal gas undergoing a reversible adiabatic change it can be shown that pvy=Ki V p -r=K2... 

Path B- C The system undergoes a reversible adiabatic change that does work on the surroundings and reduces the system temperature to Tq. 

Path D- A The system undergoes a reversible adiabatic change in which work is done on the system, the temperature remrns to Th, and the system returns to its initial state to complete the cycle. 

A reversible adiabatic expansion of an ideal gas has a zero entropy change, and an irreversible adiabatic expansion of the same gas from the same initial state to the same final volume has a positive entropy change. This statement may seem to be inconsistent with the statement that 5 is a thermodynamic property. The resolution of the discrepancy is that the two changes do not constitute the same change of state the final temperature of the reversible adiabatic expansion is lower than the final temperature of the irreversible adiabatic expansion (as in path 2 in Fig. 6.7). 

The fact that a gas can be cooled (/xJT > 0) or warmed (/zJT < 0) by merely expanding under adiabatic (adiabatic conditions, A U =w, so the work performed by the gas in reversible adiabatic expansion must be compensated by the change AU in internal energy, that is, by a temperature change (since heat capacity is nonzero). 

Notice for the reversible adiabatic expansion considered in Example 10.16 that the temperature of sample changed from 298 K to 118 K—a very dramatic temperature decrease. Because of this significant temperature decrease, the final volume of the gas is much smaller than if the expansion were carried out isothermally, where the temperature would remain at 298 K. For a reversible isothermal expansion at 298 K from P] = 10.0 atm and Vj = 12.2 L to P2 = 1.00 atm, the final volume is... 

Reversible adiabatic compression in which the gas temperature changes from T, the temperature of the cold reservoir, to T, the temperature of the hot reservoir. Since this is an adiabatic process, dQ = 0, and, from the first law, -dW = dU. The work done on the gas in this step is, therefore,... 

By measuring the temperature change and the specific volume change accompanying a small pressure change in a reversible adiabatic process, one can evaluate the derivative... 

By measuring the temperature change accompanying a differential volume change in a free expansion across a valve and separately in a reversible adiabatic expansion, the two derivatives cT/dV)H and [cT/dV)s can be experimentally evaluated. 

The first two of these equations relate to changes in state at constant entropy, that is, adiabatic, reversible changes in state. The derivative (dT/dV)s represents the rate of change of temperature with volume in a reversible adiabatic transformation. We shall not be much concerned with Eqs. (10.23) and (10.24). 

Isentropic process Any process that takes place without a change of entropy. The quantity of heat transferred, 5Q, in a reversible process is proportional to the change in entropy, 5S, i.e. 5Q= 7SS, where T is the thermodynamic temperature. Therefore, a reversible adiabatic process is isentropic, i.e. when 5Q equals zero, 5S also equals zero. 

The temperature-entropy diagram in fig. 1 clearly shows that fluids of low molar specific heat, e.g. water, can only partly be evaporated or condensed by reversible adiabatic processes. Fluids of high molar specific heat, e.g. perfluoro-n-hexane (C6F14), however, make complete adiabatic phase changes feasible. Physically, the difference between... 

---