A vessel containing 39.5 cm^3 of helium gas at 25°C and 106kPa 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 7.7cm^3 of ethanol, what was the final temperature of the helium?

Question

A vessel containing 39.5 cm^3 of helium gas at 25°C and 106kPa 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 7.7cm^3 of ethanol, what was the final temperature of the helium?

Answer 1

assuming ideal gas, we can use the formula for constant pressure:

V1/T1 = V2/T2
where
V = volume
T = temperature (in Kelvin)
note that to convert to Kelvin, K = C + 273
substituting,
39.5 / (25 + 273) = 7.7 / T2
now solve for T2. subtract 273 to convert it to deg C units

Answer 2

To find the final temperature of the helium gas, we can use the ideal gas law equation:

PV = nRT

Where:
P = pressure of the gas (in Pa)
V = volume of the gas (in m^3)
n = number of moles of gas
R = ideal gas constant (8.314 J/(mol*K))
T = temperature of the gas (in Kelvin)

First, let's convert the given volumes to cubic meters (m^3) and the given pressure to Pascals (Pa):

Given volume of helium gas = 39.5 cm^3 = 39.5 × 10^(-6) m^3
Given volume of ethanol = 7.7 cm^3 = 7.7 × 10^(-6) m^3
Given pressure = 106 kPa = 106 × 10^3 Pa

Now, we need to find the number of moles of helium gas by rearranging the ideal gas law equation:

n = PV / RT

Rearranging the equation:

n = (P × V) / (R × T)

Since the helium gas and ethanol both experience the same pressure, the number of moles of helium before and after the ethanol is forced in remains the same. Therefore, we have:

(P × V) / (R × T_initial) = (P × (V + ΔV)) / (R × T_final)

Where:
ΔV = change in volume of the gas (equal to the volume of ethanol forced in)

Substituting the given values:

(106 × 10^3 × (39.5 × 10^(-6))) / (8.314 × T_initial) = (106 × 10^3 × (39.5 × 10^(-6) + 7.7 × 10^(-6))) / (8.314 × T_final)

Simplifying the equation:

T_initial / T_final = (39.5 + 7.7) / 39.5

T_initial / T_final = 47.2 / 39.5

Now, solving for T_final:

T_final = (39.5 × T_initial) / 47.2

Since the initial temperature of the helium gas is 25°C, we need to convert it to Kelvin by adding 273.15:

T_initial = 25 + 273.15 = 298.15 K

Substituting the value of T_initial:

T_final = (39.5 × 298.15) / 47.2

T_final ≈ 250.9 K

Therefore, the final temperature of the helium gas is approximately 250.9 Kelvin.