Exact Differential for a Perfect Gas

A perfect gas is a hypothetical gas that obeys the ideal gas law at all conditions of temperature and pressure. The ideal gas law is given by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. In this context, we need to find the exact differential among the given options.

Option [a]: (P/RT)dT R.dP

Option [b]: RPdT RTdP

Option [c]: RP(dT/T) RT(dP/P)

Option [d]: RdT RT(dP/P)

Exact Differential

An exact differential is a mathematical term that describes a function whose total differential is equal to the sum of partial derivatives of its variables. In thermodynamics, an exact differential represents a property of a system that is independent of the path taken to reach a particular state.

Solution

The exact differential among the given options can be determined by checking whether the two mixed partial derivatives are equal. The mixed partial derivatives are calculated as follows:

Option [a]: (∂(P/RT)/∂P)T = 0 and (∂R/∂T)P = 0

Option [b]: (∂(RP)/∂T)P = R and (∂(RT)/∂P)T = R

Option [c]: (∂(RP)/∂P)T = R/T and (∂(RT)/∂T)P = R/P

Option [d]: (∂R/∂T)P = 0 and (∂(RT)/∂P)T = R

Therefore, the exact differential is given by option [b]: RPdT RTdP.

Conclusion

In conclusion, the exact differential for a perfect gas is RPdT RTdP, which represents the change in pressure and temperature of the gas. It is important to note that the exact differential is a property of a system that is independent of the path taken to reach a particular state.