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An HCl molecule has rotational, translat...

An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is `vec(v), m` is its mass and `k_(s)` is Bolzmann constant, then its temperature will be `(mv^(2))/(nk_(B))` , where n is __________ .

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To solve the problem, we need to derive the expression for the temperature of an HCl molecule in terms of its root mean square (RMS) velocity, mass, and Boltzmann's constant. ### Step-by-Step Solution: 1. **Understanding RMS Velocity**: The root mean square (RMS) velocity \( V_{\text{rms}} \) of gas molecules can be expressed using the formula: \[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \] where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas. 2. **Using Boltzmann's Constant**: We can also express the RMS velocity in terms of Boltzmann's constant \( k_B \): \[ V_{\text{rms}} = \sqrt{\frac{3k_B T}{m}} \] where \( m \) is the mass of a single molecule of the gas. 3. **Squaring the RMS Velocity**: Squaring both sides of the equation gives us: \[ V_{\text{rms}}^2 = \frac{3k_B T}{m} \] 4. **Rearranging for Temperature**: Rearranging the equation to solve for temperature \( T \) yields: \[ T = \frac{m V_{\text{rms}}^2}{3 k_B} \] 5. **Identifying the Value of \( n \)**: The problem states that the temperature can also be expressed as: \[ T = \frac{m V^2}{n k_B} \] By comparing the two expressions for temperature, we can see that: \[ n = 3 \] ### Final Answer: The value of \( n \) is \( 3 \).

To solve the problem, we need to derive the expression for the temperature of an HCl molecule in terms of its root mean square (RMS) velocity, mass, and Boltzmann's constant. ### Step-by-Step Solution: 1. **Understanding RMS Velocity**: The root mean square (RMS) velocity \( V_{\text{rms}} \) of gas molecules can be expressed using the formula: \[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} ...
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