`((delH)/(delP))_(T)` for an ideal gas is equal to:

Options

A. zero
B. `(DeltaVRT)/(P)`
C. `(PDeltaV)/(T)`
D. `nRDeltaT`

Answer

To find the value of \(\left(\frac{\partial H}{\partial P}\right)_{T}\) for an ideal gas, we can follow these steps:

### Step 1: Start with the differential form of enthalpy
The differential form of enthalpy \(H\) is given by:
\[ dH = TdS + VdP \]
where \(T\) is temperature, \(S\) is entropy, \(V\) is volume, and \(P\) is pressure.

### Step 2: Rearranging for \(\left(\frac{\partial H}{\partial P}\right)_{T}\)
To find \(\left(\frac{\partial H}{\partial P}\right)_{T}\), we divide the entire equation by \(dP\):
\[ \frac{dH}{dP} = T\frac{dS}{dP} + V \]
At constant temperature, \(T\) is a constant.

### Step 3: Use Maxwell's relation
From thermodynamic relations, we know that:
\[ \left(\frac{\partial S}{\partial P}\right)_{T} = \left(\frac{\partial V}{\partial T}\right)_{P} \]
This means we can substitute \(\frac{dS}{dP}\) in our equation:
\[ \frac{dH}{dP} = T\left(\frac{\partial V}{\partial T}\right)_{P} + V \]

### Step 4: Find \(\left(\frac{\partial V}{\partial T}\right)_{P}\) for an ideal gas
For an ideal gas, we know that the equation of state is given by:
\[ PV = nRT \]
Differentiating volume \(V\) with respect to temperature \(T\) at constant pressure \(P\):
\[ V = \frac{nRT}{P} \]
Taking the derivative:
\[ \left(\frac{\partial V}{\partial T}\right)_{P} = \frac{nR}{P} \]

### Step 5: Substitute back into the equation
Now substituting this back into our equation:
\[ \frac{dH}{dP} = T\left(\frac{nR}{P}\right) + V \]

### Step 6: Substitute \(V\) for an ideal gas
From the ideal gas law, we also know:
\[ V = \frac{nRT}{P} \]
Substituting this into our equation:
\[ \frac{dH}{dP} = T\left(\frac{nR}{P}\right) + \frac{nRT}{P} \]
Combining the terms:
\[ \frac{dH}{dP} = \frac{nRT}{P} + \frac{nRT}{P} = 0 \]

### Final Result
Thus, we find that:
\[ \left(\frac{\partial H}{\partial P}\right)_{T} = 0 \]