The adiabatic elasticity of hydrogen gas `(gamma=1.4)` at `NTP`

Options

A. `1xx10^5(N)/(m^2)`
B. `1xx10^8(N)/(m^2)`
C. `1.4(N)/(m^2)`
D. `1.4xx10^5(N)/(m^2)`

Explanation

To find the adiabatic elasticity of hydrogen gas at Normal Temperature and Pressure (NTP), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Adiabatic Elasticity**:
Adiabatic elasticity refers to the elasticity of a gas when it undergoes an adiabatic process, meaning no heat is exchanged with the surroundings. For a gas, this can be defined using the specific heat ratio (γ).

2. **Using the Relationship**:
The relationship for an adiabatic process is given by:
\[ pV^\gamma = \text{constant} \]
where \( p \) is the pressure, \( V \) is the volume, and \( \gamma \) (gamma) is the specific heat ratio.

3. **Differentiating the Equation**:
To find the adiabatic elasticity, we differentiate both sides of the equation with respect to volume \( V \):
\[ \frac{d}{dV}(pV^\gamma) = 0 \]
This leads to:
\[ p \cdot \gamma \cdot V^{\gamma - 1} + V^\gamma \frac{dp}{dV} = 0 \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ p \cdot \gamma = -\frac{dp}{dV} \cdot V \]
This indicates that the change in pressure with respect to volume is related to the pressure and the specific heat ratio.

5. **Defining Adiabatic Elasticity**:
The adiabatic elasticity \( E_a \) can be defined as:
\[ E_a = p \cdot \gamma \]

6. **Substituting Values**:
At NTP, the pressure \( p \) is approximately \( 10^5 \) Pascal (or \( 100,000 \) Pa). Given \( \gamma = 1.4 \):
\[ E_a = 10^5 \, \text{Pa} \cdot 1.4 = 1.4 \times 10^5 \, \text{Pa} \]

7. **Final Result**:
Therefore, the adiabatic elasticity of hydrogen gas at NTP is:
\[ E_a = 1.4 \times 10^5 \, \text{N/m}^2 \]

### Summary:
The adiabatic elasticity of hydrogen gas at NTP is \( 1.4 \times 10^5 \, \text{N/m}^2 \).

---