First, we need to figure out which gas law is applicable here. The question states that temperature and moles are constant. This means that we are dealing with Boyle’s law, which states that pressure is inversely proportional to volume. Inversely proportional means that the pressure decreases as volume increases and vice versa. Note that the relationship is still linear (change in one variable causes a proportional change in the other variable), but the two variables have a negative correlation. Positive correlation means increasing or decreasing a variable would also increase or decrease the other variable, respectively. 

Recall that Charles’ law defines the relationship between temperature and volume at constant pressure and constant moles. The law states that the two variables are directly proportional to each other. This means that increasing or decreasing temperature will have the same effect on volume. This makes sense because increasing the temperature increases the kinetic energy of the particles. This makes it easier for particles to move away from each other and expand, which results in an increase in volume.

To solve this question we need to use the ideal gas law equation:

Above,  is pressure in ,  is volume in liters,  is moles,  is , and  is temperature in Kelvins. The question gives us pressure, volume, and temperature; therefore, we can solve for . First, we need to convert temperature from Celsius to Kelvins.

Rearrange the ideal gas law and solve for :

The question states that you place one gram of gas in the container. Since we know the moles, we can solve for the molecular weight of the gas and figure out its identity from the molecular weight.

The molecular weight of oxygen gas,  is ; therefore, the unknown gas must be oxygen.

The argon gas, as the problem states, is under constant temperature with varying volume and pressure, so we need to use Boyle's Law:

We are given the initial pressure, the initial volume, and the final pressure, and need to solve for the final volume. Therefore, we can rearrange Boyle's law to be as follows: 

Plug in known values and solve.

The methane gas, as the problem states, is under constant temperature with varying volume and pressure, so we need to use Boyle's Law:

We are given the initial pressure, the initial volume, and the final volume and need to solve for the final pressure. Therefore, we can rearrange Boyle's law to be as follows:

Plug in known values and solve.

Condition I is true. Pressure and volume are inversely proportional. The ideal gas law with volume can be rederived to show this:

We introduce an expression for moles:

Where  is moles,  is mass, and  is molar mass in 

Rearranging (1) to solve for  and plugging into the following:

We get:

 (2)

Finally, we use the following for density, where  is volume, and plug into (2)

The masses cancel out and we have the ideal gas law expressed with volume and moles.

Condition II is false. It is clear from the equation that pressure and density are directly proportional.

Condition III is true because it is clear from the equation that temperature and pressure are directly proportional.

Condition IV is true because density and temperature are on the same side of the equation in the numerator, so the must be inversely proportional.

Condition V is false because the ideal gas constant and molar mass can be rearranged to be on opposite sides of the equation and in the numerator.

Gases deviate from ideal conditions at low temperature and high pressure. This because the postulates of the kinetic molecular theory of gasses ignore the volume of the molecules and all interactions between gas molecules. However, neither are true for real gasses. As the temperature increases, the kinetic energy of the particles can overcome the intermolecular forces of attraction or repulsion between the molecules. At high pressures, and subsequently low volume, the distance between molecules becomes shorter, and therefore intermolecular forces become significant. So, ideal conditions are when the distance between molecules is great, and the energy each molecule has is much greater in relative magnitude to the intermolecular forces. This occurs when the pressure is low, and the temperature is high.

All the statements say similar things, but only one correctly identifies the resulting change to volume and pressure. Let's break things down by the correction.

Correction 1:

The correction decreases the volume, since it is subtracting  from the overall volume. As n (or the number of molecules) goes up, volume will go down.

Correction 2:

Since we are subtracting  from the overall pressure, we are correcting for molecular attraction by decreasing pressure. We are doing this with a term  that gets larger when  increases and and gets larger as  decreases.

We can use the boiling point elevation equation in order to find the new boiling point for the solution.

The change in temperature will be equal to the boiling point elevation constant multiplied by the molality of the solution multiplied by the van't Hoff factor. The van't Hoff factor is a variable that incorporates how many ions the added solute will dissolve into when in solution. Sodium chloride will dissolve into two ions, so the van't Hoff factor is 2.

This means that the boiling point of the water has been elevated by 5.2 degrees, meaning that the boiling point of the solution is 105.2 degrees celsius.

We can use the freezing point depression equation in order to determine the molar mass of the unknown solute.