A vessel containing 39.5 cm3 of helium gas at 25°C and 106 kPa was inverted and placed in cold ethanol. As the gas contracted, ethanol was forced into the vessel to maintain the same pressure of helium. If this required 18.8 cm3 of ethanol, what was the final temperature of the helium?
(V1/T1) = (V2/T2)
V1 = 39.5
T1 = 298
V2 = 39.5-18.8
T2 = ?
2859.88k
To solve this problem, we can use the ideal gas law, which states that 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.
First, let's calculate the initial number of moles of helium using the given initial conditions:
P1 = 106 kPa
V1 = 39.5 cm^3
T1 = 25°C = 25 + 273.15 = 298.15 K (converted to Kelvin)
Using the ideal gas law, we can rearrange the equation to solve for n:
n = (P1 * V1) / (R * T1)
Now, let's calculate the number of moles of helium:
n1 = (106 * 39.5) / (8.314 * 298.15)
Next, when the vessel is inverted and ethanol is forced into the vessel, the total number of moles of gas remains the same. Therefore, n1 = n2.
Now, we know that V2 = V1 + volume of ethanol added:
V2 = V1 + 18.8 cm^3
Rearranging this equation, we can solve for V1:
V1 = V2 - 18.8
Substituting n2 = n1, V2 = V1 + 18.8, and rearranging the ideal gas law equation, we can solve for the final temperature, T2:
T2 = (P2 * V2) / (n2 * R)
Since the pressure remains the same (106 kPa), we can calculate the final temperature:
T2 = (106 * (V1 + 18.8)) / (n1 * R)
Now, let's calculate the final temperature of the helium gas.
To determine the final temperature of the helium gas, we can use the combined gas law equation, which relates the initial and final pressure, volume, and temperature of a gas sample. The equation is as follows:
(P1 * V1) / T1 = (P2 * V2) / T2
Where:
P1 = initial pressure
V1 = initial volume
T1 = initial temperature
P2 = final pressure
V2 = final volume
T2 = final temperature
Given:
P1 = 106 kPa
V1 = 39.5 cm3
T1 = 25°C
V2 = V1 + volume of ethanol = 39.5 cm3 + 18.8 cm3 = 58.3 cm3
P2 = P1 (since the pressure is maintained)
We need to find T2 (final temperature of the helium gas)
Now, let's substitute the given values into the equation:
(106 kPa * 39.5 cm3) / (25°C) = (106 kPa * 58.3 cm3) / T2
Next, we can simplify the equation by cross-multiplying:
(106 kPa * 39.5 cm3 * T2) = (106 kPa * 58.3 cm3 * 25°C)
Now, divide both sides of the equation by (106 kPa * 39.5 cm3) to solve for T2:
T2 = (106 kPa * 58.3 cm3 * 25°C) / (106 kPa * 39.5 cm3)
Canceling out the common units (kPa and cm3), we get:
T2 = (58.3 cm3 * 25°C) / (39.5 cm3)
T2 = 36.962°C
Therefore, the final temperature of the helium gas is approximately 36.962°C.