An unsealed 750 mL beaker contains 500 mL of very dense gas at 25.0 degrees C and 1.10 atm. The gas is much denser than the surrounding air, so it does not disperse into the air.
(a) At constant pressure, what would the temperature have to reach for the gas to fill the beaker?
(b) At constant temperature, what would the pressure have to reach for the gas to fill the beaker? (assume the liquid will always stay in liquid form)
P1V1/T1=P2V2/T2
a) p1=P2, solve for T2
b) t1=t2, solve for P2
temps in kelvins.
To answer these questions, we can use the combined gas law equation, which relates the temperature, pressure, and volume of a gas sample. The formula is as follows:
(P1 * V1) / T1 = (P2 * V2) / T2
Where:
P1 and P2 are the initial and final pressures,
V1 and V2 are the initial and final volumes, and
T1 and T2 are the initial and final temperatures.
Let's solve the questions step-by-step:
(a) To find the temperature at which the gas will fill the beaker at constant pressure, we need to calculate the final volume of the gas.
Given:
Initial volume, V1 = 500 mL
Final volume, V2 = 750 mL
Initial pressure, P1 = 1.10 atm
Substituting these values into the formula, we get:
(1.10 atm * 500 mL) / T1 = (1.10 atm * 750 mL) / T2
Cancelling out the "atm" units and rearranging the equation, we have:
500 mL / T1 = 750 mL / T2
Cross-multiplying, we get:
500 mL * T2 = 750 mL * T1
Dividing both sides by 500 mL, we get:
T2 = (750 mL * T1) / 500 mL
Simplifying further, we have:
T2 = 1.5 * T1
So, the temperature at which the gas will fill the beaker at constant pressure is 1.5 times the initial temperature.
(b) To find the pressure at which the gas will fill the beaker at constant temperature, we need to calculate the final pressure of the gas.
Given:
Initial volume, V1 = 500 mL
Initial temperature, T1 = 25.0 degrees C
Final volume, V2 = 750 mL
Substituting these values into the formula, we have:
(1.10 atm * 500 mL) / (25.0 degrees C) = (P2 * 750 mL) / (25.0 degrees C)
Cancelling out the "mL" and "degrees C" units, we get:
44 atm = (P2 * 750 mL) / 25
Cross-multiplying, we get:
44 atm * 25 = P2 * 750 mL
Dividing both sides by 750 mL, we have:
P2 = (44 atm * 25) / 750 mL
Simplifying further, we get:
P2 = 1.47 atm
So, the pressure at which the gas will fill the beaker at constant temperature is approximately 1.47 atm.
To answer these questions, we need to apply the ideal gas law, which states that the product of pressure (P) and volume (V) is proportional to the number of moles (n) of gas and its temperature (T):
PV = nRT
where R is the ideal gas constant.
Let's consider each question separately:
(a) At constant pressure, what would the temperature have to reach for the gas to fill the beaker?
In this case, the pressure remains constant, and we want the gas to fill the entire beaker. To accomplish this, the volume of the gas needs to increase. According to the ideal gas law, when the volume increases, the temperature also needs to increase to maintain a constant pressure.
So, to find the temperature at which the gas will fill the beaker, we can set up an equation using the initial conditions:
P_initial * V_initial = P_final * V_final
Here, the initial conditions are:
- P_initial = 1.10 atm (given)
- V_initial = 500 mL = 0.5 L (given)
And the final conditions (where the gas fills the beaker completely):
- P_final = 1.10 atm (same as initial pressure since it's constant)
- V_final = 750 mL = 0.75 L (volume of the beaker)
Rearranging the equation, we get:
V_final = (P_initial * V_initial) / P_final
Substituting the given values, we have:
0.75 L = (1.10 atm * 0.5 L) / 1.10 atm
Simplifying, we find:
0.75 L = 0.5 L
Since the volume of the beaker is larger than the initial volume, the gas will fill the beaker at any temperature above the initial temperature of 25.0 degrees C (298.15 K).
(b) At constant temperature, what would the pressure have to reach for the gas to fill the beaker?
In this case, the temperature remains constant, and we want the gas to fill the beaker. Again, using the ideal gas law, when the volume increases, the pressure needs to decrease to maintain a constant temperature.
So, to find the pressure at which the gas will fill the beaker, we can set up an equation using the initial conditions:
P_initial * V_initial = P_final * V_final
As before, the initial conditions are:
- P_initial = 1.10 atm (given)
- V_initial = 500 mL = 0.5 L (given)
And the final conditions (where the gas fills the beaker completely):
- P_final = ? (what we want to find)
- V_final = 750 mL = 0.75 L (volume of the beaker)
Rearranging the equation, we get:
P_final = (P_initial * V_initial) / V_final
Substituting the given values, we have:
P_final = (1.10 atm * 0.5 L) / 0.75 L
Simplifying, we find:
P_final = 0.733 atm
Therefore, to fill the beaker with the gas at a constant temperature of 25.0 degrees C (298.15 K), the pressure needs to decrease to approximately 0.733 atm.