Total pressure in a chamber calculation
Dalton's Law is only true for ideal compounds, which theoretically have no intermolecular forces between their molecules. Those forces become more and more significant when increasing the concentration of any compound, because all molecules become packed up closer.

This will lower the pressure, because if any molecule is about to hit the container wall, it gets pulled backwards $-$ and thus slowed $-$ due to the attraction of the other molecules behind it. But for high temperature this effect is negligible, because they can't be slowed down so easily. 

Now lets come to the calculation using Kay's rule:

Since you only have one liquid component, you don't need Roult's Law to calculate the vapor pressure for a mixture of liquids. 

1.) You need to find the values for the critical temperatures and critical pressures for your components (gases and water vapor). 

2.) You need to know the relative amounts of each substance:
$$y_i = \frac{n_i}{n_m}$$

$n_i$... amount of a specific component [mol] 

$n_m$... total amount of substance of the mixture [mol] 

3.) Calculate the pseudo-critical temperature and pressure:
$$T_c' = y_1 \times T_{c1} + y_2 \times T_{c2} + y_3 \times T_{c3} $$

$$P_c' = y_1\times P_{c1} +y_2\times P_{c2} +y_3\times P_{c3}$$

$T_c$... critical temperature of a specific component [K] 

$P_c$... critical pressure of a specific component [Pa] 

4.) calculate the pseudo-reduced values for T and the volume:
$$T_r = T / T_c'$$
$$v_r' = V / (R \times T_c' / P_c')$$

$T$... temperature in the container

$V$... volume of the gas phase [m$^3$]

$R$... universal gas constant = 8.314 J/(K mol)

5.) Now you need to look up $v_r$ and $T_r$ in a compressibility chart to get the value for a specific $Z$ factor, where their lines intersect. 

6.) And finally calculate the pressure: $$P = ZRT / V$$

This will be more accurate than if you would use the standard way for ideal gases, especially for low temperatures

References:
real gases

calculation

compressibility chart examples