Question
Read the passage given below and answer the following questions from (i) to (v). Pressure of an Ideal Gas: according to kinetic theory of gases pressure is given by $\text{P}=\frac{1}{3}\text{ nmv}^2$
Where, n is number of molecules per unit volume, m is mass and $v^2$ is mean squared speed. Though we choose the container to be a cube, the shape of the vessel really is immaterial. The average kinetic energy of a molecule is proportional to the absolute temperature of the gas; it is independent of pressure, volume or the nature of the ideal gas. This is a fundamental result relating temperature, a macroscopic measurable parameter of a gas (a thermodynamic variable as it is called) to a molecular quantity, namely the average kinetic energy of a molecule. The two domains are connected by the Boltzmann constant and given by $E = k_bT$. Where kb is Boltzmann constant having value of $1.38 \times 10^{-23}$ joule per Kelvin. We have seen that in thermal equilibrium at absolute temperature T, for each translational mode of motion, the average energy is $\frac{1}{2}\text{K}_\text{b}\text{t}$. The most elegant principle of classical statistical mechanics (first proved by Maxwell) states that this is so for each mode of energy: translational, rotational and vibrational. That is, in equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to $\frac{1}{2}\text{K}_\text{b}\text{t}$. This is known as the law of equipartition of energy. Accordingly, each translational and rotational degree of freedom of a molecule contributes $\frac{1}{2}\text{K}_\text{b}\text{t}$ to the energy, while each vibrational frequency contributes $2\times\frac{1}{2}\text{Kb T}=\text{K}_\text{b}\text{T}$ since a vibrational mode has both kinetic and potential energy modes.
Where, n is number of molecules per unit volume, m is mass and $v^2$ is mean squared speed. Though we choose the container to be a cube, the shape of the vessel really is immaterial. The average kinetic energy of a molecule is proportional to the absolute temperature of the gas; it is independent of pressure, volume or the nature of the ideal gas. This is a fundamental result relating temperature, a macroscopic measurable parameter of a gas (a thermodynamic variable as it is called) to a molecular quantity, namely the average kinetic energy of a molecule. The two domains are connected by the Boltzmann constant and given by $E = k_bT$. Where kb is Boltzmann constant having value of $1.38 \times 10^{-23}$ joule per Kelvin. We have seen that in thermal equilibrium at absolute temperature T, for each translational mode of motion, the average energy is $\frac{1}{2}\text{K}_\text{b}\text{t}$. The most elegant principle of classical statistical mechanics (first proved by Maxwell) states that this is so for each mode of energy: translational, rotational and vibrational. That is, in equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to $\frac{1}{2}\text{K}_\text{b}\text{t}$. This is known as the law of equipartition of energy. Accordingly, each translational and rotational degree of freedom of a molecule contributes $\frac{1}{2}\text{K}_\text{b}\text{t}$ to the energy, while each vibrational frequency contributes $2\times\frac{1}{2}\text{Kb T}=\text{K}_\text{b}\text{T}$ since a vibrational mode has both kinetic and potential energy modes.
- Boltzmann constant has value of:
- 1.38 × 10 - 23 joule per Kelvin.
- 1.38 × 10 - 28 joule per Kelvin.
- 1.38 × 10 - 30 joule per Kelvin.
- None of these.
- SI unit of Boltzmann constant is given by:
- Joules per meter
- Joules per Kelvin
- Joules per Newton
- None of these
- According to kinetic theory give formula for pressure of idea gas.
- According to kinetic theory what is average kinetic energy of molecules of ideal gas?
- What is law of equipartition of energy?
