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

Options

A. Zero
B. `1xx10^5 N//m^2`
C. `1.4 xx 10^5 N//m^2`
D. `2.75 xx 10^5 N//m^2`

Answer

To find the adiabatic elasticity of hydrogen gas at NTP, we can follow these steps:

### Step 1: Understand the Concept of Adiabatic Elasticity
Adiabatic elasticity (or bulk modulus of elasticity) is defined as the ratio of the change in pressure to the relative change in volume. Mathematically, it is expressed as:
\[ E_a = -\frac{dP}{dV} \cdot \frac{V}{P} \]
Where \(E_a\) is the adiabatic elasticity, \(P\) is the pressure, \(V\) is the volume, and \(dP\) and \(dV\) are the changes in pressure and volume, respectively.

### Step 2: Use the Relationship for Adiabatic Processes
For an adiabatic process, we have the relation:
\[ PV^\gamma = \text{constant} \]
Where \(\gamma\) (gamma) is the adiabatic index. For hydrogen gas, \(\gamma = 1.4\).

### Step 3: Relate Adiabatic Elasticity to Pressure and Volume
From the relationship of adiabatic processes, we can derive:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]
This implies:
\[ \frac{dP}{dV} = -\gamma \frac{P}{V} \]

### Step 4: Substitute into the Formula for Adiabatic Elasticity
Substituting \(\frac{dP}{dV}\) into the formula for adiabatic elasticity:
\[ E_a = -\left(-\gamma \frac{P}{V}\right) \cdot \frac{V}{P} = \gamma P \]

### Step 5: Calculate the Pressure at NTP
At Normal Temperature and Pressure (NTP), the pressure \(P\) is \(10^5 \, \text{Pa}\) (or \(100,000 \, \text{Pa}\)).

### Step 6: Calculate the Adiabatic Elasticity
Now substituting the values:
\[ E_a = \gamma P = 1.4 \times 10^5 \, \text{Pa} \]
Calculating this gives:
\[ E_a = 1.4 \times 100,000 = 140,000 \, \text{Pa} = 1.4 \times 10^5 \, \text{Pa} \]

### Conclusion
Thus, the adiabatic elasticity of hydrogen gas at NTP is:
\[ E_a = 1.4 \times 10^5 \, \text{Pa} \]

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