A certain substance has a heat vaporization of 47.97 kJ/mol.
At what Kelvin temperature will the vapor pressure be 7.50 times higher than it was at 297K?
Please help me! I don't understand heat vaporization questions
Plug these numbers into the Clausius-Clapeyron equation.
To solve this problem, you need to understand the relationship between temperature, vapor pressure, and heat vaporization.
The Clausius-Clapeyron equation relates vapor pressure (P), temperature (T), and molar heat of vaporization (ΔHvap) of a substance:
ln(P2/P1) = -ΔHvap/R * (1/T2 - 1/T1)
Where:
P2 and P1 are the vapor pressures at temperatures T2 and T1
ΔHvap is the molar heat of vaporization
R is the ideal gas constant (8.314 J/(mol*K))
T2 and T1 are the temperatures at which the vapor pressures are measured
In this case:
P1 = vapor pressure at 297K
P2 = 7.50 times P1
T1 = 297K
T2 = temperature at which the vapor pressure is 7.50 times higher
First, we need to find P2 by multiplying P1 by 7.50.
Now, we can rearrange the Clausius-Clapeyron equation to solve for T2:
ln(P2/P1) = -ΔHvap/R * (1/T2 - 1/T1)
Substituting the given values:
ln(7.50) = -ΔHvap/R * (1/T2 - 1/297)
Now, solve this equation for T2.
Step 1: Divide both sides by -ΔHvap/R
ln(7.50) / (-ΔHvap/R) = 1/T2 - 1/297
Step 2: Simplify the left side of the equation.
ln(7.50) / (-ΔHvap/R) = [(1 * 297 - 1 * T2) / (297 * T2)]
Step 3: Simplify the right side of the equation.
ln(7.50) / (-ΔHvap/R) = (297 - T2) / (297 * T2)
Step 4: Cross-multiply.
[-ΔHvap/R * ln(7.50)] * (297 * T2) = 297 - T2
Step 5: Distribute -ΔHvap/R * ln(7.50) with 297 * T2.
[-ΔHvap/R * ln(7.50)] * 297 * T2 = 297 - T2
Step 6: Move T2 terms to one side of the equation.
[-ΔHvap/R * ln(7.50)] * 297 * T2 + T2 = 297
Step 7: Factor out T2.
[T2 * ([-ΔHvap/R * ln(7.50)] * 297 + 1)] = 297
Step 8: Divide both sides by ([-ΔHvap/R * ln(7.50)] * 297 + 1).
T2 = 297 / ([-ΔHvap/R * ln(7.50)] * 297 + 1)
Now, substitute the given value for ΔHvap and R into the equation and solve for T2:
T2 = 297 / ([-47.97 × 10^3 J/mol / (8.314 J/(mol·K)) × ln(7.50)] × 297 + 1)
Using a calculator or mathematical software, calculate T2 to find the Kelvin temperature at which the vapor pressure will be 7.50 times higher than it was at 297K.