Question

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if the temperature of an ideal gas doubles, what happens to the root mean square speed?

decreases by a factor of \\(\sqrt{2}\\).

increases by a factor of 4.

increases by a factor of \\(\sqrt{2}\\).

increases a factor of 2.

what is the internal energy of two moles of a monatomic ideal gas at a temperature of 127°c?

9970 j

4990 j

1580 j

3170 j

Explanation:

First Question (Root Mean Square Speed)

Step1: Recall RMS Speed Formula

The root - mean - square (rms) speed of an ideal gas is given by the formula \(v_{rms}=\sqrt{\frac{3kT}{m}}\) (or \(v_{rms}=\sqrt{\frac{3RT}{M}}\), where \(k\) is Boltzmann's constant, \(T\) is the absolute temperature, \(m\) is the mass of a molecule, \(R\) is the gas constant, and \(M\) is the molar mass).

Step2: Analyze the Effect of Temperature Change

Let the initial temperature be \(T_1\) and the final temperature be \(T_2 = 2T_1\). The initial rms speed \(v_{rms1}=\sqrt{\frac{3kT_1}{m}}\), and the final rms speed \(v_{rms2}=\sqrt{\frac{3kT_2}{m}}=\sqrt{\frac{3k(2T_1)}{m}}\).
We can rewrite \(v_{rms2}\) as \(\sqrt{2}\times\sqrt{\frac{3kT_1}{m}}\), and since \(\sqrt{\frac{3kT_1}{m}} = v_{rms1}\), then \(v_{rms2}=\sqrt{2}v_{rms1}\). So the rms speed increases by a factor of \(\sqrt{2}\).

Step1: Recall Internal Energy Formula for Monatomic Ideal Gas

The internal energy \(U\) of a monatomic ideal gas is given by the formula \(U=\frac{3}{2}nRT\), where \(n\) is the number of moles, \(R = 8.31\space J/(mol\cdot K)\) is the gas constant, and \(T\) is the absolute temperature.

Step2: Convert Temperature to Kelvin

The given temperature is \(T = 127^{\circ}C\). To convert to Kelvin, we use \(T(K)=T(^{\circ}C)+273\). So \(T = 127 + 273=400\space K\), and \(n = 2\space mol\).

Step3: Substitute Values into the Formula

Substitute \(n = 2\), \(R = 8.31\), and \(T = 400\) into the formula \(U=\frac{3}{2}nRT\).
\(U=\frac{3}{2}\times2\times8.31\times400\)
First, \(\frac{3}{2}\times2 = 3\). Then, \(3\times8.31\times400=3\times3324 = 9972\space J\approx9970\space J\) (due to rounding of \(R\) value).

Answer:

Increases by a factor of \(\boldsymbol{\sqrt{2}}\) (the option "Increases by a factor of \(\sqrt{2}\)")

Second Question (Internal Energy of Monatomic Ideal Gas)