This is a problem about the solubility of a gas in a liquid, which is affected by pressure, temperature, and other factors. The solubility of a gas in a liquid is usually measured in terms of its mole fraction, which is the ratio of the number of moles of gas to the total number of moles (gas plus liquid).
In this case, we can use Henry's law, which states that the mole fraction of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. Mathematically, we can write:
x = k * P
where x is the mole fraction of the gas in the liquid, P is the partial pressure of the gas above the liquid, and k is the proportionality constant, which depends on the gas and the liquid.
To solve this problem, we need to use the given information to find the value of k, and then use Henry's law to find the mole fraction of gas in the second solution.
First, we can use the given information to find the value of k:
x = k * P
0.75 g of gas at 40 atm of pressure dissolves in 1.25 l of water, which has a density of about 1 g/cm^3 at 25°C. The molar mass of the gas is not given, but we can assume that it is a simple gas such as nitrogen, oxygen, or carbon dioxide, which have molar masses of about 28 g/mol, 32 g/mol, and 44 g/mol, respectively. Let's assume that the gas is nitrogen, which has a molar mass of 28 g/mol. Then, the number of moles of gas dissolved in the water is:
n = m / M = 0.75 g / 28 g/mol ≈ 0.0268 mol
The total number of moles in the solution is equal to the number of moles of gas plus the number of moles of water:
n_total = n_gas + n_water
n_total = n_gas + V_water / M_water
where V_water is the volume of water and M_water is the molar mass of water (18 g/mol).
Since the density of water is about 1 g/cm^3, we can convert the volume of water from liters to cubic centimeters:
1.25 l = 1250 cm^3
Then, we can calculate the total number of moles in the solution:
n_total = 0.0268 mol + 1250 cm^3 / 1000 cm^3/mol
n_total ≈ 0.048 mol
The mole fraction of the gas in the solution is equal to the number of moles of gas divided by the total number of moles:
x = n_gas / n_total
x = 0.0268 mol / 0.048 mol
x ≈ 0.558
Now we can use Henry's law to find the mole fraction of gas in the second solution:
x = k * P
We want to find x when P = 3.0 atm and V_water = 2.0 L. Since the temperature is the same as in the first solution, we can assume that k has the same value as before.
k = x / P
k = 0.558 / 40 atm
k ≈ 0.014
x = k * P
x = 0.014 * 3.0 atm
x = 0.042
Therefore, the mole fraction of gas in the second solution is about 0.042. To find the number of moles of gas in the second solution, we can use the same formula as before:
n_gas = x * n_total
n_gas = 0.042 * 0.048 mol
n_gas ≈ 0.002 mol
Finally, we can convert the number of moles of gas to mass using the molar mass of nitrogen:
m_gas = n_gas * M_gas
m_gas = 0.002 mol * 28 g/mol
m_gas = 0.056 g
Therefore, approximately 0.056 g of gas will dissolve in 2.0 L of water at 3.0 atm of pressure and the same temperature.