First, we understand that the mean free path of gas molecules is given by the formula:
- \(\lambda = \frac{k_{\text{B}} T}{\sqrt{2} \pi d^2 P}\)
where:
- \(\lambda\) is the mean free path,
- \(k_{\text{B}} = 1.38 \times 10^{-23}\text{ J/K}\) is the Boltzmann constant,
- \(T\) is the temperature in Kelvin,
- \(d = 5 \times 10^{-10}\text{ m}\) is the diameter of a molecule,
- \(P = 1.38 \times 10^{5}\text{ Pa}\) is the pressure.
Convert the temperature from Celsius to Kelvin:
Temperature in Kelvin: \(T = 41^{\circ}\text{C} + 273 = 314 \text{ K}\)
Substitute the given values into the formula:
\(\lambda = \frac{1.38 \times 10^{-23} \times 314}{\sqrt{2} \times \pi \times (5 \times 10^{-10})^2 \times 1.38 \times 10^{5}}\)
Simplify the calculation:
- \((5 \times 10^{-10})^2 = 25 \times 10^{-20}\)
- \(\text{Denominator} = \sqrt{2} \times \pi \times 25 \times 10^{-20} \times 1.38 \times 10^{5}\)
- Substitute: \(\lambda = \frac{1.38 \times 10^{-23} \times 314}{\sqrt{2} \times \pi \times 25 \times 10^{-20} \times 1.38 \times 10^{5}}\)
- Simplifying further: \(\lambda \approx 10\sqrt{2} \times 10^{-8}\) m
Thus, the mean free path of the molecule is \(10\sqrt{2} \times 10^{-8}\) m.
Similar Questions
- Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is $32 \times 10^{18}$ /s then collision frequency in gas A is _________ /s.
- An insulated cylinder of volume $60 \text{ cm}^3$ is filled with a gas at $27^\circ\text{C}$ and 2 atmospheric pressure. Then the gas is compressed making the final volume as $20 \text{ cm}^3$ while allowing the temperature to rise to $77^\circ\text{C}$. The final pressure is _________ atmospheric pressure.
- A brass wire of length 2 m and radius 1 mm at $27^\circ\text{C}$ is held taut between two rigid supports. Initially it was cooled to a temperature of $-43^\circ\text{C}$ creating a tension $T$ in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to $1.4T$, is ______ $^\circ\text{C}$.
- A gas of certain mass filled in a closed cylinder at a pressure of 3.23 kPa has temperature $50^\circ\text{C}$. The gas is now heated to double its temperature. The modified pressure is ______ Pa.
10 mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from $P_1$ to $P_2$ is $\alpha \text{ Joule}$ ($P_1 = 21.7 \text{ Pa}$ and $P_2 = 30 \text{ Pa}, C_v = 21 \text{ J/K.mol}, R = 8.3 \text{ J/mol.K}$). The value of $\alpha$ is _______.
- When $300 \text{ J}$ of heat given to an ideal gas with $C_p = \frac{7}{2} R$ its temperature raises from $20^\circ\text{C}$ to $50^\circ\text{C}$ keeping its volume constant. The mass of the gas is (approximately) _______ g. ($R = 8.314 \text{ J/mol.K}$)
A thermodynamic system is taken through the cyclic process ABC as shown in the figure. The total work done by the system during the cycle $ABC$ is _________ $\text{J}$.
- A certain gas is isothermally compressed to $\left(\frac{1}{3}\right)^{\text{rd}}$ of its initial volume ($V_o = 3$ litre) by applying required pressure. If the bulk modulus of the gas is $3 \times 10^5 \text{ N/m}^2$, the magnitude of work done on the gas is _________ J.
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Statement I: Change in internal energy of a system containing $n$ mole of ideal gas can be written as $\Delta U = n C_v (T_f - T_i) = \frac{nR}{\gamma - 1}(T_f - T_i)$, where $\gamma = \frac{C_p}{C_v}$, $T_i = \text{initial temperature}$, $T_f = \text{final temperature}$.
Statement II: Relation between degree of freedom $f$ and $\gamma (= C_p / C_v)$ is $\left(\gamma = 1 + \frac{2}{f}\right)$
Choose the correct answer from the options given below- Consider the following statements:
A. Zeroth law of thermodynamics gives concept of temperature
B. First law of thermodynamics gives concept of internal energy
C. In isothermal expansion of ideal gas, $\Delta Q \neq \Delta W$
D. Product of intensive and extensive variables is extensive
E. The ratio of any extensive variable to mass will be an extensive variable
Choose the correct combination of statements from the options given below:
Important Questions from Heat and Thermodynamics
- Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is $32 \times 10^{18}$ /s then collision frequency in gas A is _________ /s.
- An insulated cylinder of volume $60 \text{ cm}^3$ is filled with a gas at $27^\circ\text{C}$ and 2 atmospheric pressure. Then the gas is compressed making the final volume as $20 \text{ cm}^3$ while allowing the temperature to rise to $77^\circ\text{C}$. The final pressure is _________ atmospheric pressure.
- A brass wire of length 2 m and radius 1 mm at $27^\circ\text{C}$ is held taut between two rigid supports. Initially it was cooled to a temperature of $-43^\circ\text{C}$ creating a tension $T$ in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to $1.4T$, is ______ $^\circ\text{C}$.
- A gas of certain mass filled in a closed cylinder at a pressure of 3.23 kPa has temperature $50^\circ\text{C}$. The gas is now heated to double its temperature. The modified pressure is ______ Pa.
10 mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from $P_1$ to $P_2$ is $\alpha \text{ Joule}$ ($P_1 = 21.7 \text{ Pa}$ and $P_2 = 30 \text{ Pa}, C_v = 21 \text{ J/K.mol}, R = 8.3 \text{ J/mol.K}$). The value of $\alpha$ is _______.
