Coefficient of isothermal elasticity `E_(theta)` and coefficient of adiabatic elasticity `E_(phi)` are related by `(gamma = C_(p) //C_(v))`

Options

A. `E_(theta) = gamma E_(phi)`
B. `E_(phi) = gamma E_(theta)`
C. `E_(theta) = gamma//E_(phi)`
D. `E_(theta) = gamma^(2)E_(phi)`

Answer

To solve the relationship between the coefficient of isothermal elasticity \( E_{\theta} \) and the coefficient of adiabatic elasticity \( E_{\phi} \), we can follow these steps:

### Step 1: Understand the Definitions
- The coefficient of isothermal elasticity \( E_{\theta} \) is defined as the ratio of the change in pressure to the change in volume at constant temperature.
- The coefficient of adiabatic elasticity \( E_{\phi} \) is defined as the ratio of the change in pressure to the change in volume during an adiabatic process.

### Step 2: Establish the Relationship
From thermodynamics, we know that the relationship between the coefficients can be expressed as:
\[ E_{\phi} = \gamma \cdot E_{\theta} \]
where \( \gamma \) (gamma) is the ratio of specific heats, defined as:
\[ \gamma = \frac{C_p}{C_v} \]
Here, \( C_p \) is the specific heat at constant pressure, and \( C_v \) is the specific heat at constant volume.

### Step 3: Derive the Coefficients
1. **Isothermal Process**: For an isothermal process, the coefficient of isothermal elasticity \( E_{\theta} \) can be expressed as:
\[ E_{\theta} = -\frac{P}{V} \frac{dV}{dP} \]
where \( P \) is the pressure and \( V \) is the volume.

2. **Adiabatic Process**: For an adiabatic process, the coefficient of adiabatic elasticity \( E_{\phi} \) can be expressed as:
\[ E_{\phi} = -\frac{dP}{dV} \cdot V \]

### Step 4: Substitute and Relate
Using the relationship \( E_{\phi} = \gamma \cdot E_{\theta} \):
- Substitute the expressions for \( E_{\theta} \) and \( E_{\phi} \):
\[ E_{\phi} = \gamma \left(-\frac{P}{V} \frac{dV}{dP}\right) \]
This shows that \( E_{\phi} \) is directly proportional to \( E_{\theta} \) scaled by the factor \( \gamma \).

### Conclusion
Thus, we conclude that the coefficient of adiabatic elasticity \( E_{\phi} \) is related to the coefficient of isothermal elasticity \( E_{\theta} \) by the equation:
\[ E_{\phi} = \gamma \cdot E_{\theta} \]

### Final Answer
The relationship between the coefficients is given by \( E_{\phi} = \gamma \cdot E_{\theta} \).

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