Find ratio of quantities of water and of argon in airtight container

Question

I am working on my EJU exam past papers (2013) and got stuck at this question. Knowledge of high school or first year university chemistry is assumed. I honestly do not know where I should start.

When argon and water were placed in an airtight container with a constant volume and the gas mixture was kept at 373 K, the pressure of the gas mixture became P₁ with a portion of water remained as liquid. When the whole was heated to 403 K, all the water was vaporized and the pressure became P₂. From (1) to (6) below choose the correct formula representing the ratio, water/argon, of the quantities (in mol) of water and of argon in the container. Assume that the vapor pressure of water at 373 K is P_v, that argon is an ideal gas not dissolving in water, and that the volume of water before heating to 403 K can be neglected.

Please note that answer key provided is (1). I tried to think that we could form an equation from total moles of gases equate to the sum of total moles of water and argon at 403 K since all are vaporized. Then use the ideal gas law to find the total amount of gases at 403 K. However, I cannot connect these pieces and information given at 373 K together. Could anyone give me a solution to this question or at least a hint so I can continue working on it?

Answer 1

You should use the ideal gas law to convert between $n_i$ (amount of gas) and $p/T$. For each temperature and pressure pair you can write $$\begin{align}n_{g,1}=\frac{p_1V}{RT_1}\\n_{g,2}=\frac{p_2V}{RT_2}\end{align}$$ where $$\begin{align}n_{g,1}&=n_{g,H_2O}+n_{Ar}\\n_{H_2O}&=n_{g,H_2O}+n_{l,H_2O}\\n_{g,2}&=n_{H_2O}+n_{Ar}\end{align}$$ You also know that for each individual gas the partial pressure is proportional to the number of moles (Dalton's law): $$n_i=\frac{p_iV}{RT}$$ The "trick" is that at $T_1=\pu{373K}$ $$p_{Ar} = p_1 - p_V$$ where $p_V$ is given as the partial pressure of water at that temperature. You should be able to show that $$\frac{n_{H_2O}}{n_{Ar}}=\frac{n_{g,2}}{n_{Ar}}-1$$ and making appropriate substitutions it should not be difficult to arrive at the answer.