Partial Pressure of a Hg-N2 System

[wmrunner24]

Homework Statement

At 1 atm of pressure a volume of 22 liters of N₂ gas is passed in a closed system over a boat containing Hg liquid at 100°C. The flow of N₂ is slow to allow the gas to become saturated with mercury. At 20°C and 1 atm, the nitrogen was found to contain 0.0647g of Hg. Calculate the vapor pressure of Hg at 100°C.

Homework Equations

Raoult's Law (Ideal Solution):
P_Hg = X_HgP°_Hg

Ideal Gas Law:

PV = nRT

The Attempt at a Solution

My endgame here is to use Raoult's Law to calculate the partial pressure of mercury. To do that, though, I will need the mole fraction X_Hg and the pressure the mercury would exert if it were alone in the container P°_Hg. As far as the latter goes, I suspect I can use the ideal gas law to calculate the pressure of mercury when nitrogen is not involved.

This is where I run into problems. I'm not sure how to synthesize the information given into the information I need. I could calculate a mole fraction for the system at 20°C, again via the ideal gas law, but I'm not sure how I can relate this to either P°_Hg or X_Hg for the same system at 100°C.

Any insight anyone can provide would be incredibly helpful and greatly appreciated.

 

[Chestermiller]

I think that the problem statement meant to say that 0.0647 gm of mercury condensed out of the N2 when the gas was cooled to 20 C. I also think it is safe to say that this was virtually all the Hg that had evaporated into the 22 liters of N2 (assuming that the Hg vapor pressure to 20 C is much lower than the vapor pressure at 100C). How many moles of Hg condensed? How many moles of N2 were in the mixture (it should be on the order of 1 mole)? What was the mole fraction of Hg in the gas phase when it got saturated with Hg at 100C? Treating the gas phase as an ideal gas, what was the vapor pressure of Hg at 100C?

 

[wmrunner24]

Okay. I think this was the missing link I needed. Let me make sure though:

n_Hg = m_Hg/M_Hg, where M_Hg is the atomic mass of mercury, 200.59 g/mol

n_Hg = 3.23 x 10^-4 mol

The number of moles of nitrogen can be calculated using the ideal gas law:

n_N2 = $\frac{PV}{RT}$ = 0.718 mol, with P = 1 atm, V = 22 L, T = 373 K.

I think you were right about the number of moles being ≈1, but because these are not exactly STP conditions, we get a number that is slightly lower.

From here, the mole fraction X_Hg is trivial to calculate. The pressure of Hg in its own container P°_Hg is given again by the ideal gas law, using the above n_Hg, T, and V from previous calculations. Then the partial pressure P_Hg is just a matter of using Raoult's Law:

P_Hg = X_HgP°_Hg

I got P_Hg = 2.015 x 10^-7 atm. Is this correct?

Thank you so much for your assistance with this problem.

 

[Chestermiller]

Okay. I think this was the missing link I needed. Let me make sure though:

n_Hg = m_Hg/M_Hg, where M_Hg is the atomic mass of mercury, 200.59 g/mol

n_Hg = 3.23 x 10^-4 mol

The number of moles of nitrogen can be calculated using the ideal gas law:

n_N2 = $\frac{PV}{RT}$ = 0.718 mol, with P = 1 atm, V = 22 L, T = 373 K.

I think you were right about the number of moles being ≈1, but because these are not exactly STP conditions, we get a number that is slightly lower.

From here, the mole fraction X_Hg is trivial to calculate. The pressure of Hg in its own container P°_Hg is given again by the ideal gas law, using the above n_Hg, T, and V from previous calculations. Then the partial pressure P_Hg is just a matter of using Raoult's Law:

P_Hg = X_HgP°_Hg

I got P_Hg = 2.015 x 10^-7 atm. Is this correct?

Thank you so much for your assistance with this problem.

I'm not quite sure what you did after getting the number of moles of the Hg vapor that were mixed with the 0.718 moles of N2 in the gas at 100 C, but Raoult's law was not the thing to use. N2 is essentially insoluble in the liquid Hg, so the mole fraction of Hg in the liquid phase is 1.0 (i.e., virtually pure Hg). But, getting back to the gas phase, you can calculate the mole fractions of N2 and Hg in the gas phase at 100 C (since you know how many moles of each is present), and you know the total pressure, so you can calculate the partial pressure of each gas (equal to the mole fraction times the total pressure). Because the liquid Hg was pure, its partial pressure for the gas mixture at 100 C is equal to its equilibrium vapor pressure.

 

[rezeew]

I would first start by identifying the key variables and equations that are relevant to this problem. In this case, we are dealing with a closed system containing both nitrogen and mercury at different temperatures and pressures. The key equations that can help us solve this problem are Raoult's Law and the Ideal Gas Law.

Raoult's Law states that the vapor pressure of a component in a solution is equal to the product of its mole fraction in the solution and its vapor pressure when it is pure. This can be written as:

PHg = XHgP°Hg

Where PHg is the partial pressure of mercury, XHg is the mole fraction of mercury in the solution, and P°Hg is the vapor pressure of mercury when it is pure.

The Ideal Gas Law, on the other hand, relates the pressure, volume, temperature, and number of moles of a gas in a system. It can be written as:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

Using these equations, we can solve for the vapor pressure of mercury at 100°C. First, we need to calculate the mole fraction of mercury in the solution at 20°C. We can do this by using the ideal gas law and assuming that the volume of the system remains constant:

nHg = PV/RT = (1 atm)(22 L)/(0.0821 L atm/mol K)(293 K) = 0.888 mol

We know that the total number of moles in the system is 0.888 mol, so the mole fraction of mercury can be calculated as:

XHg = nHg/ntotal = 0.888 mol/0.888 mol = 1

This means that at 20°C, the mole fraction of mercury in the solution is 1. This makes sense since the system is closed and only contains mercury and nitrogen.

Next, we need to calculate the vapor pressure of mercury at 20°C. This can be done by rearranging Raoult's Law:

P°Hg = PHg/XHg = (0.0647 g/0.888 mol)(0.0821 L atm/mol K)(293 K) = 0.143 atm

Now, we can use this value of P°Hg and Ra