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 17.8 cm3 of ethanol, what was the final temperature of the helium?
To find the final temperature of the helium gas, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure in Pascal (Pa)
V = volume in cubic meters (m^3)
n = number of moles of gas
R = ideal gas constant (8.314 J/(mol*K))
T = temperature in Kelvin (K)
First, let's convert the given values to the appropriate units:
- The initial volume of the helium gas is 39.5 cm^3, so we need to convert it to m^3 by dividing by 100 (1 cm^3 = 0.000001 m^3). Therefore, the initial volume (V1) is 0.000395 m^3.
- The initial pressure (P1) is given as 106 kPa. We need to convert it to Pa by multiplying by 1000 (1 kPa = 1000 Pa). Therefore, P1 = 106,000 Pa.
- The final volume of the helium-ethanol mixture is given as 17.8 cm^3, so we need to convert it to m^3. Therefore, the final volume (V2) is 0.000178 m^3.
Now, we can rewrite the ideal gas law equation as:
P1V1 = P2V2
Substituting the given values:
(106,000 Pa)(0.000395 m^3) = (P2)(0.000178 m^3)
Simplifying further:
(106,000)(0.000395) = P2(0.000178)
42.07 = P2(0.000178)
Now, we need to find the final pressure (P2). Rearranging the equation:
P2 = 42.07 / 0.000178
P2 ≈ 236,404.5 Pa
Next, we need to find the final temperature (T2) in Kelvin. Since the number of moles (n) and the gas constant (R) remain constant, we can rewrite the equation as:
PV = nRT
Therefore:
P1V1 / T1 = P2V2 / T2
Substituting the given values:
(106,000 Pa)(0.000395 m^3) / (25°C + 273.15 K) = (236,404.5 Pa)(0.000178 m^3) / T2
Simplifying further:
(106,000)(0.000395) / (25 + 273.15) = (236,404.5)(0.000178) / T2
0.04214 ≈ 0.04201 / T2
Now, we can solve for T2 by isolating it on one side of the equation:
T2 ≈ 0.04201 / 0.04214
T2 ≈ 0.9989
Therefore, the final temperature of the helium gas is approximately 0.9989 Kelvin.