(Solved): CPCV=T(TP)V(TV)P a) Show that for the ideal gas this yields the we ...

Expert Answer

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To show that for an ideal gas, the relation Cp = Cv + nR holds using the given equation   , we can apply the ideal gas law and the definition of specific heat capacities.
Starting with the given equation:
  
We know that for an ideal gas, the specific heat capacities are defined as follows:    (heat capacity at constant pressure)    (heat capacity at constant volume)
Using the first law of thermodynamics, we have:    (change in heat energy = change in internal energy + work done)
For an ideal gas, the change in internal energy (dU) only depends on the change in temperature (dT). Therefore, we can write:   

Substituting this into the equation for heat capacity at constant pressure (Cp), we have: Cp = (?Q/?T)p = dU/dT + P(dV/dT)p
Now, let's consider the ideal gas law: PV = nRT
Taking the partial derivative of this equation with respect to time (t) at constant volume (V), we get: V(?P/?t)v = nR
Substituting this into the expression for Cp, we have: Cp = dU/dT + P(dV/dT)p = Cv + (nR/V)(dV/dT)p
Since nR/V = P (using the ideal gas law), we have: Cp = Cv + P(dV/dT)p
Therefore, for an ideal gas, the relation Cp = Cv + nR holds true.
Now, to express the relation in terms of the number of moles (n) and the ideal gas constant (R), we can use the ideal gas law: PV = nRT
Solving for P, we have: P = nRT/V
Substituting this back into the equation for Cp, we get: Cp = Cv + P(dV/dT)p = Cv + (nRT/V)(dV/dT)p
Since the partial derivative (?V/?T)p represents the change in volume with respect to temperature at constant pressure, it is related to the thermal expansion coefficient (?) of the gas: (?V/?T)p = V?
Therefore, we have: Cp = Cv + P(dV/dT)p = Cv + P(V?) = Cv + nRT?/V = Cv + nR?
So, the relation for an ideal gas can be expressed as: Cp = Cv + nR
Thus, we have shown that for an ideal gas, Cp = Cv + nR, which is the well-known relationship between the specific heat capacities.