Slope of isotherm for a gas (having `gamma = (5)/(3)`) is `3 xx 10^(5) N//m^(2)` . If the same gas is undergoing adiabatic change then adiabatic elasticity at that instant is

Options

A. `3 xx 10^(5) N//m^2`
B. `5 xx 10^(5) N//m^2`
C. `6 xx 10^(5) N//m^2`
D. `10 xx 10^(5) N//m^2`

Answer

To solve the problem step by step, we need to find the adiabatic modulus of elasticity for the gas given the slope of the isotherm and the value of gamma (γ).

### Step 1: Understand the given information
We are given:
- The slope of the isotherm (dP/dV) = 3 × 10^5 N/m²
- The adiabatic constant (γ) = 5/3

### Step 2: Relate the slope of the isotherm to the isothermal modulus of elasticity
The slope of the isotherm is related to the isothermal modulus of elasticity (E_iso) by the equation:
\[ E_{\text{iso}} = -\frac{dP}{dV} \]
Since we are interested in the modulus, we take the absolute value:
\[ E_{\text{iso}} = \left| -\frac{dP}{dV} \right| = \frac{dP}{dV} = 3 \times 10^5 \, \text{N/m}^2 \]

### Step 3: Use the relationship for adiabatic modulus of elasticity
The adiabatic modulus of elasticity (E_adiabatic) is given by the formula:
\[ E_{\text{adiabatic}} = \gamma \frac{P}{V} \]
We need to find the value of P/V. From the isothermal process, we know that:
\[ \frac{dP}{dV} = -\frac{P}{V} \]
Thus,
\[ \frac{P}{V} = -\frac{dP}{dV} \]

### Step 4: Substitute the slope into the equation
Since we have the slope (dP/dV) from the isotherm:
\[ \frac{P}{V} = -\left(3 \times 10^5\right) \]
Now substituting this into the equation for adiabatic modulus of elasticity:
\[ E_{\text{adiabatic}} = \gamma \left(-\frac{dP}{dV}\right) \]
Substituting the values:
\[ E_{\text{adiabatic}} = \frac{5}{3} \times \left(3 \times 10^5\right) \]

### Step 5: Calculate the adiabatic modulus of elasticity
Now, simplifying the expression:
\[ E_{\text{adiabatic}} = \frac{5}{3} \times 3 \times 10^5 = 5 \times 10^5 \, \text{N/m}^2 \]

### Final Answer
The adiabatic modulus of elasticity at that instant is:
\[ E_{\text{adiabatic}} = 5 \times 10^5 \, \text{N/m}^2 \]

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