Represent equation of an adiabatic process in terms of (i) `T` and `V` (ii) `P` and `T`.

Answer

To represent the equation of an adiabatic process in terms of (i) Temperature (T) and Volume (V) and (ii) Pressure (P) and Temperature (T), we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat transfer to or from the system, which means \( \Delta Q = 0 \). According to the first law of thermodynamics:
\[ \Delta Q = dU + dW \]
This implies that:
\[ dU = -dW \]

### Step 2: Use the Ideal Gas Law
For an ideal gas, the equation of state is given by:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

### Step 3: Derive the Relation Between P and V
For an adiabatic process, the relation between pressure and volume is given by:
\[ PV^\gamma = \text{constant} \]
where \( \gamma = \frac{C_p}{C_v} \) is the heat capacity ratio.

### Step 4: Derive the Relation Between T and V
From the ideal gas law, we can express \( P \) in terms of \( T \) and \( V \):
\[ P = \frac{nRT}{V} \]
Substituting this into the adiabatic equation:
\[ \left(\frac{nRT}{V}\right)V^\gamma = k \]
This simplifies to:
\[ nRTV^{\gamma - 1} = k \]
Rearranging gives:
\[ TV^{\gamma - 1} = \frac{k}{nR} \]
Thus, the relation between \( T \) and \( V \) for an adiabatic process is:
\[ TV^{\gamma - 1} = \text{constant} \]

### Step 5: Derive the Relation Between P and T
Now, we want to express the adiabatic condition in terms of \( P \) and \( T \). From the ideal gas law, we can express \( V \) as:
\[ V = \frac{nRT}{P} \]
Substituting this into the adiabatic equation:
\[ P\left(\frac{nRT}{P}\right)^\gamma = k \]
This simplifies to:
\[ P \frac{(nRT)^\gamma}{P^\gamma} = k \]
Rearranging gives:
\[ P^{1 - \gamma} T^\gamma = \text{constant} \]
Thus, the relation between \( P \) and \( T \) for an adiabatic process is:
\[ P T^{\frac{\gamma}{\gamma - 1}} = \text{constant} \]

### Final Relations:
1. **In terms of T and V**:
\[ TV^{\gamma - 1} = \text{constant} \]
2. **In terms of P and T**:
\[ P T^{\frac{\gamma}{\gamma - 1}} = \text{constant} \]