Pressure-temperature relationship for an ideal gas undergoing adiabatic change is (`gamma = C_p//C_v`)

Options

A. `PT^(gamma)` = constant
B. `PT^(-1+gamma)` = constant
C. `p^(gamma-1)T^(gamma)`= constant
D. `p^(1-gamma)T^(gamma)`= constant

Explanation

To derive the pressure-temperature relationship for an ideal gas undergoing an adiabatic change, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. For an ideal gas undergoing an adiabatic change, the relationship between pressure (P), volume (V), and temperature (T) can be described by the equation:
\[ PV^\gamma = \text{constant} \]
where \(\gamma = \frac{C_p}{C_v}\) (the ratio of specific heats).

### Step 2: Use the Ideal Gas Law
The ideal gas law is given by:
\[ PV = nRT \]
where \(n\) is the number of moles and \(R\) is the universal gas constant. We can rearrange this equation to express volume (V) in terms of pressure (P) and temperature (T):
\[ V = \frac{nRT}{P} \]

### Step 3: Substitute Volume into the Adiabatic Equation
Now, we substitute the expression for volume (V) from the ideal gas law into the adiabatic equation:
\[ P \left( \frac{nRT}{P} \right)^\gamma = \text{constant} \]

### Step 4: Simplify the Equation
This simplifies to:
\[ P \cdot \frac{(nRT)^\gamma}{P^\gamma} = \text{constant} \]
which can be rearranged to:
\[ P^{1 - \gamma} \cdot (nRT)^\gamma = \text{constant} \]

### Step 5: Isolate the Terms
Rearranging gives us:
\[ P^{1 - \gamma} \cdot n^\gamma \cdot R^\gamma \cdot T^\gamma = \text{constant} \]
We can denote the constant as \(C\), leading to:
\[ P^{1 - \gamma} \cdot T^\gamma = \frac{C}{n^\gamma R^\gamma} \]

### Step 6: Final Relationship
Thus, we can express the relationship between pressure and temperature for an ideal gas undergoing an adiabatic process as:
\[ P^{\gamma - 1} \cdot T^{\gamma} = \text{constant} \]

### Conclusion
This is the required pressure-temperature relationship for an ideal gas undergoing an adiabatic change.

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