Animated Solution for Physics - Thermodynamics: A mixture of hydrogen and oxygen has volume $500{\text{ cm}}^3$, temperature $300\text{ K}$, pressure $400\text{ kPa}$ and mass $0.76\text{ g}$. The ratio of masses of oxygen to hydrogen will be

Imagine a sealed container holding a mixture of hydrogen ($H_2$) and oxygen ($O_2$) gases. We are given the macroscopic properties of this mixture: its volume $V=500{\text{ cm}}^3$, pressure $p=400\text{ kPa}$, temperature $T=300\text{ K}$, and total mass $m_{\text{total}}=0.76\text{ g}$.

Our goal is to peek inside this macroscopic data and figure out the exact ratio of the masses of oxygen to hydrogen. To do this, we need to bridge the gap between the macroscopic world (pressure, volume, temperature) and the microscopic world (moles and molecules).

Ideal Gas Law

To find the individual masses, we first need to know the total number of moles in the mixture. Since the mixture behaves as an ideal gas, we can use the ideal gas equation:

Rearranging for the total number of moles $n$, we get:

Let's carefully substitute the given values. We must convert pressure to Pascals and volume to cubic meters to maintain SI units.

Calculating the numerator gives $200$, and the denominator is roughly $2494.2$. Dividing these gives us approximately $0.08\text{ mol}$. So, the sum of moles of hydrogen ($n_1$) and oxygen ($n_2$) is:

Now, let's use the second piece of information: the total mass. The total mass is simply the sum of the mass of hydrogen and the mass of oxygen. We can express mass as the number of moles multiplied by the molar mass:

The molar mass of hydrogen gas ($H_2$) is $2\text{ g/mol}$, and for oxygen gas ($O_2$), it is $32\text{ g/mol}$. Substituting these, we get:

We can simplify this by dividing the entire equation by $2$:

Now we have a neat system of two linear equations. Let's subtract our first equation from this new equation:

The $n_1$ terms cancel out beautifully, leaving:

This simplifies to $n_2=0.02\text{ mol}$. Consequently, substituting this back into equation (1), $n_1$ must be $0.06\text{ mol}$.

Finally, we need the ratio of the mass of oxygen to the mass of hydrogen. We plug in our calculated moles and the molar masses:

Substituting the values:

Multiplying numerator and denominator by 100 to remove decimals: