Question

two containers are filled with he(g) under the conditions shown in the diagram. which of the following statements about the average speeds of the atoms in the containers is correct?
a the he(g) atoms in container x have the same average speed as the he(g) atoms in container y because the volumes are the same.
b the he(g) atoms in container x have the same average speed as the he(g) atoms in container y because the pressures are the same.

Explanation:

Step1: Recall the formula for average speed

The average speed of gas molecules is given by \(v_{avg}=\sqrt{\frac{8RT}{\pi M}}\), where \(R\) is the gas constant, \(T\) is the temperature, and \(M\) is the molar mass.

Step2: Analyze the effect of temperature

Since \(M\) (for \(He\), \(M = 4\space g/mol\)) and \(R\) are constants, \(v_{avg}\propto\sqrt{T}\). Container \(X\) has \(T = 298\space K\) and container \(Y\) has \(T=333\space K\). As \(T_Y>T_X\), the average speed of \(He\) atoms in \(Y\) is greater than in \(X\).

Step3: Analyze options A and B

Options A and B claim that the average speeds are the same. But from the formula \(v_{avg}\propto\sqrt{T}\), volume (\(V\)) and pressure (\(P\)) do not directly affect the average speed (when considering the formula for average speed of gas molecules). The key factor is temperature.

Answer:

Neither option A nor option B is correct. The average speed of gas molecules depends on temperature (\(v_{avg}\propto\sqrt{T}\)), not on volume or pressure. Since \(T_X = 298\space K\) and \(T_Y=333\space K\), the \(He\) atoms in container \(Y\) have a higher average speed than those in container \(X\).