A vessel containing 39.5 cm^3 of helium gas at 25 degrees celsius 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 tihs required 7.7 cm^3 of ethanol, what was the final temperature of the helium?
(Pressure doesn't matter here either, since it remains constant. So you're left with V/T=V/T, which is Charles' Law.)
PV/nT = PV/nT
UNITS:
------
P: atm (convert kPa to atm)
V: L (convert cm^3 to L)
n: mol (moles don't matter in this problem)
T: K (convert C to K)
( They need to fit with the ideal gas constant [R]'s units: [L*atm/K*mol] )
To find the final temperature of the helium, we can use the ideal gas law equation: 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 convert the given volume of helium and ethanol to liters, as the ideal gas constant is commonly used with SI units.
Volume of helium = 39.5 cm^3 = 39.5/1000 = 0.0395 L
Volume of ethanol = 7.7 cm^3 = 7.7/1000 = 0.0077 L
Since the ethanol is forced into the vessel at the same pressure as the helium, the change in volume for helium is equal to the volume of ethanol added. Therefore, the final volume of the helium is:
Final volume of helium = Initial volume of helium + Volume of ethanol
= 0.0395 L + 0.0077 L
= 0.0472 L
The number of moles of helium remains constant since the container is closed and no gas escapes. Therefore, the equation can be written as:
(Pressure of helium) * (Initial volume of helium) = (Pressure of helium) * (Final volume of helium) = nRT
Rearranging the equation to solve for temperature (T), we have:
T = (Pressure of helium) * (Final volume of helium) / (nR)
Now, let's substitute the given values:
Pressure of helium = 106 kPa
Final volume of helium = 0.0472 L (which we calculated)
Number of moles of helium = unknown
Ideal gas constant (R) = 8.314 J/(mol·K)
To find the number of moles of helium, we can use the ideal gas law equation with the initial conditions of helium.
(Pressure of helium) * (Initial volume of helium) = (Number of moles of helium) * (Ideal gas constant) * (Initial temperature of helium)
Rearranging the equation to solve for the number of moles of helium (n), we have:
n = (Pressure of helium) * (Initial volume of helium) / (Ideal gas constant * Initial temperature of helium)
Given:
Pressure of helium = 106 kPa
Initial volume of helium = 0.0395 L (which we calculated)
Ideal gas constant (R) = 8.314 J/(mol·K)
Initial temperature of helium = 25 degrees Celsius = 25 + 273.15 = 298.15 K
Now, let's substitute these values to find the number of moles of helium:
n = (106 kPa) * (0.0395 L) / (8.314 J/(mol·K) * 298.15 K)
After performing the calculations, we find that the number of moles of helium is approximately 0.001741 mol.
Now, let's substitute the values into the equation to find the final temperature of the helium:
T = (106 kPa) * (0.0472 L) / (0.001741 mol * 8.314 J/(mol·K))
After performing the calculations, we find that the final temperature of the helium is approximately 181.3 K.
Therefore, the final temperature of the helium is approximately 181.3 Kelvin.