High School

The relationship between the vapor pressure of a liquid and temperature can be expressed by the Clausius-Clapeyron equation. Ethanol (\(C_2H_5OH\)) has a normal boiling point of 78.3°C with \(\Delta H_{vap} = 39.3 \text{ kJ/mol}\).

What is the vapor pressure of ethanol at 50.0°C?

A. 118 torr
B. 234 torr
C. 354 torr
D. 485 torr
E. 670 torr

Answer :

Final answer:

The vapor pressure of ethanol at 50.0°C is approximately 117.8 torr.

Explanation:

To find the vapor pressure of ethanol at 50.0°C, we can use the Clausius-Clapeyron equation:

ln(P2/P1) = -(AHvap/R) * (1/T2 - 1/T1)

Where:

  • P1 is the vapor pressure at the boiling point (78.3°C)
  • P2 is the vapor pressure at the given temperature (50.0°C)
  • AHvap is the enthalpy of vaporization (39.3 kJ/mol)
  • R is the ideal gas constant (8.314 J/(mol·K))
  • T1 is the boiling point temperature in Kelvin (78.3 + 273.15)
  • T2 is the given temperature in Kelvin (50.0 + 273.15)

Substituting the values into the equation:

ln(P2/118) = -(39.3 * 10^3 / 8.314) * (1/(50.0 + 273.15) - 1/(78.3 + 273.15))

Simplifying the equation:

ln(P2/118) = -4.732 * (0.00365 - 0.00265)

ln(P2/118) = -4.732 * 0.001

ln(P2/118) = -0.004732

Using the natural logarithm properties, we can solve for P2:

P2/118 = e^(-0.004732)

P2 = 118 * e^(-0.004732)

P2 ≈ 117.8 torr

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Answer:

The Clausius-Clapeyron equation relates the vapor pressure of a substance to its temperature and enthalpy of vaporization:

ln(P2/P1) = (ΔHvap / R) * (1/T1 - 1/T2)

Where:

P1 = Vapor pressure at temperature T1

P2 = Vapor pressure at temperature T2

ΔHvap = Enthalpy of vaporization

R = Gas constant (8.314 J/mol·K or 0.0821 L·atm/mol·K)

T1 = Temperature in Kelvin at T1

T2 = Temperature in Kelvin at T2

Given that the normal boiling point of ethanol is 78.3°C, we can convert this to Kelvin:

T1 = 78.3°C + 273.15 = 351.45 K

The enthalpy of vaporization (ΔHvap) is given as 39.3 kJ/mol, which needs to be converted to J/mol:

ΔHvap = 39.3 kJ/mol * 1000 J/kJ = 39300 J/mol

Now we are given T2 = 50.0°C, which we'll convert to Kelvin:

T2 = 50.0°C + 273.15 = 323.15 K

Now we can plug these values into the Clausius-Clapeyron equation:

ln(P2/P1) = (39300 J/mol / (8.314 J/mol·K)) * (1/351.45 K - 1/323.15 K)

Solving for ln(P2/P1), we get:

ln(P2/P1) = 5.7878

Now, we need to solve for P2/P1:

P2/P1 = e^(5.7878)

P2/P1 ≈ 323.59

Now, we want to find P2, the vapor pressure of ethanol at 50.0°C (T2):

P2 = P1 * (P2/P1)

P2 = P1 * 323.59

Given that the vapor pressure of ethanol at its normal boiling point (P1) is 1 atm (760 torr), we can calculate P2:

P2 = 760 torr * 323.59 ≈ 245869.4 torr

However, this result seems unusually high for the vapor pressure of ethanol at 50.0°C. There might be a mistake or incorrect data in the problem. Please double-check the values provided or the calculation steps.

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