Question 808: Thermodynamics & KTG

Question

A sample of oxygen gas and a sample of hydrogen gas both have the same mass, same volume and the same pressure. The ratio of their absolute temperature is (Molecular wt. of $\mathrm{O}_2 \& \mathrm{H}_2$ is 32 and 2 respectively)

Accepted Answer

The ideal gas law is given by:
$$ PV = nRT $$
Where:
$ P $ is the pressure,
$ V $ is the volume,
$ n $ is the number of moles,
$ R $ is the ideal gas constant,
$ T $ is the absolute temperature.
For the two gases, since the mass, volume, and pressure are the same for both oxygen ($\mathrm{O}_2$) and hydrogen ($\mathrm{H}_2$), we can set up the following expressions for each gas:
For oxygen:
$$ P V = n_{\mathrm{O}_2} R T_{\mathrm{O}_2} $$
For hydrogen:
$$ P V = n_{\mathrm{H}_2} R T_{\mathrm{H}_2} $$
Since $ PV $ is the same for both gases, we have:
$$ n_{\mathrm{O}_2} R T_{\mathrm{O}_2} = n_{\mathrm{H}_2} R T_{\mathrm{H}_2} $$
This simplifies to:
$$ n_{\mathrm{O}_2} T_{\mathrm{O}_2} = n_{\mathrm{H}_2} T_{\mathrm{H}_2} $$
The number of moles ($ n $) is given by:
$$ n = \frac{	ext{mass}}{	ext{molar mass}} $$
Let $ m $ be the mass of each gas. Then:
For oxygen:
$$ n_{\mathrm{O}_2} = \frac{m}{32} $$
For hydrogen:
$$ n_{\mathrm{H}_2} = \frac{m}{2} $$
Substitute into the equation:
$$ \frac{m}{32} T_{\mathrm{O}_2} = \frac{m}{2} T_{\mathrm{H}_2} $$
Cancel $ m $ from both sides:
$$ \frac{1}{32} T_{\mathrm{O}_2} = \frac{1}{2} T_{\mathrm{H}_2} $$
Cross-multiply to find the ratio of temperatures:
$$ 2 T_{\mathrm{O}_2} = 32 T_{\mathrm{H}_2} $$
Simplify:
$$ T_{\mathrm{O}_2} = 16 T_{\mathrm{H}_2} $$
Thus, the ratio of the absolute temperatures is:
$$ T_{\mathrm{O}_2} : T_{\mathrm{H}_2} = 16 : 1 $$
The correct option is
Option C: $16 : 1$
.

Answer

$1: 4$

Answer

$1: 8$

Answer

$16: 1$

Answer

$12: 1$

Thermodynamics & KTG

A sample of oxygen gas and a sample of hydrogen gas both have the same mass, same volume and the same pressure. The ratio of their absolute temperature is (Molecular wt. of \mathrm{O}_2 \& \mathrm{H}_2 is 32 and 2 respectively)

Choose the correct answer: