Given an expression relating the Celsius scale and Kelvin scale. Write an expression for the root mean square speed of a gas.

Here are the solutions to the problems.

1. Given an expression relating the Celsius scale and Kelvin scale. Write an expression for the root mean square speed of a gas.

• Relationship between Celsius and Kelvin scales: The temperature in Kelvin ($T_K$) is related to the temperature in Celsius ($T_C$) by the following expression:

  $$T_K = T_C + 273.15$$

• Expression for the root mean square (RMS) speed of a gas: The root mean square speed ($v_{rms}$) of gas molecules is given by the formula:

  $$v_{rms} = \sqrt{\frac{3RT}{M}}$$

  where:

  • $R$ is the ideal gas constant ($8.314 \text{ J mol}^{-1} \text{ K}^{-1}$)
  • $T$ is the absolute temperature in Kelvin
  • $M$ is the molar mass of the gas in $\text{kg mol}^{-1}$

2. Calculate the root mean square speed of hydrogen molecule at 300K (molar mass of hydrogen is 2.016 g/mol, R = 8.314 J mol⁻¹ K⁻¹).