Question 265: Thermodynamics & KTG

Question

The rms speed of oxygen molecule in a vessel at particular temperature is $$\left(1+\frac{5}{x}\right)^{\frac{1}{2}} v$$, where $$v$$ is the average speed of the molecule. The value of $$x$$ will be:
$$\left(\right.$$ Take $$\left.\pi=\frac{22}{7}\right)$$

Accepted Answer

The relationship between the root-mean-square (rms) speed ($$v_{rms}$$) and the average speed ($$v_{avg}$$) of molecules in a gas can be found using the Maxwell-Boltzmann distribution. The rms speed and average speed are related as follows:
$$v_{rms} = \sqrt{\frac{3RT}{M}}$$

$$v_{avg} = \sqrt{\frac{8RT}{\pi M}}$$
Where:
$$R$$ is the ideal gas constant
$$T$$ is the temperature in Kelvin
$$M$$ is the molar mass of the gas
$$\pi$$ is the mathematical constant pi
In this problem, the rms speed of the oxygen molecule is given by:
$$v_{rms} = \left(1+\frac{5}{x}\right)^{\frac{1}{2}} v_{avg}$$
Now, let's divide the expression for $$v_{rms}$$ by the expression for $$v_{avg}$$:
$$\frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}} = \left(1+\frac{5}{x}\right)^{\frac{1}{2}}$$
By simplifying the expression, we get:
$$\frac{v_{rms}}{v_{avg}} = \frac{\sqrt{3}}{\sqrt{\frac{8}{\pi}}} = \left(1+\frac{5}{x}\right)^{\frac{1}{2}}$$
Square both sides of the equation:
$$\frac{3}{\frac{8}{\pi}} = 1 + \frac{5}{x}$$
Now we will substitute the provided value of $$\pi = \frac{22}{7}$$:
$$\frac{3}{\frac{8}{\frac{22}{7}}} = 1 + \frac{5}{x}$$
By simplifying the expression, we get:
$$\frac{3 \cdot \frac{22}{7}}{8} = 1 + \frac{5}{x}$$
Now let's solve for $$x$$:
$$\frac{66}{56} - 1 = \frac{5}{x}$$

$$\frac{10}{56} = \frac{5}{x}$$
Multiplying both sides by $$x$$:
$$\frac{10}{56}x = 5$$
Finally, solving for $$x$$:
$$x = \frac{5 \cdot 56}{10} = 28$$
So, the value of $$x$$ is $$\boxed{28}$$.

Answer

4

Answer

8

Answer

28

Answer

27

Thermodynamics & KTG

The rms speed of oxygen molecule in a vessel at particular temperature is $\left(1+\frac{5}{x}\right)^{\frac{1}{2}} v, where v is the average speed of the molecule. The value of x will be: \left(\right. Take \left.\pi=\frac{22}{7}\right)

Choose the correct answer: