Calculate the change in the boiling point of water (in K) when the pressure is increased from 1 to 10 atm.
Given - Specific volume of steam = 1677 cc / gm
Latent heat of steam = 540 cal / gm
boiling point of water at one atm pressure = 100⁰C = 373 k
and pressure of one atm = 10⁶ dyne / cm²
Correct answer is '248.07'. Can you explain this answer?

Question

Calculate the change in the boiling point of water (in K) when the pressure is increased from 1 to 10 atm.
Given - Specific volume of steam = 1677 cc / gm
Latent heat of steam = 540 cal / gm
boiling point of water at one atm pressure = 100⁰C = 373 k
and pressure of one atm = 10⁶ dyne / cm²

Correct answer is '248.07'. Can you explain this answer?

Answer

Calculation of Change in Boiling Point of Water

Let's calculate the change in boiling point of water when the pressure is increased from 1 to 10 atm.

Given:
Specific volume of steam = 1677 cc/gm
Latent heat of steam = 540 cal/gm
Boiling point of water at 1 atm pressure = 100°C = 373 K
Pressure of 1 atm = 106 dyne/cm^2

Step 1: Conversion of Pressure
We need to convert the pressure from atm to dyne/cm^2.

1 atm = 106 dyne/cm^2

So, the pressure of 10 atm will be:
10 atm * 106 dyne/cm^2 = 1060 dyne/cm^2

Step 2: Calculation of Change in Boiling Point
To calculate the change in boiling point, we can use the Clausius-Clapeyron equation:

ΔT = (ΔHvap/R) * (1/T1 - 1/T2)

Where:
ΔT = Change in boiling point
ΔHvap = Latent heat of vaporization
R = Gas constant
T1 = Initial boiling point temperature
T2 = Final boiling point temperature

Given values:
ΔHvap = 540 cal/gm
T1 = 373 K
T2 = T1 + ΔT (unknown)
R = 8.314 J/(mol⋅K) (universal gas constant)

Step 3: Conversion of Units
The given latent heat of vaporization is in cal/gm, but the gas constant is in J/(mol⋅K). We need to convert ΔHvap to J/mol.

1 cal = 4.184 J
1 gm = 0.001 kg

So, the conversion factor will be:
(540 cal/gm) * (4.184 J/cal) / (0.001 kg/1000 gm) = 2257.376 J/kg

Step 4: Calculation of Change in Boiling Point
Now, we can substitute the values into the Clausius-Clapeyron equation:

ΔT = (ΔHvap/R) * (1/T1 - 1/T2)

ΔT = (2257.376 J/kg / (8.314 J/(mol⋅K))) * (1/373 K - 1/T2)

Simplifying the equation:
ΔT = (271.54/mol⋅K) * (1/373 K - 1/T2)

Step 5: Solving the Equation
We can solve the equation by substituting the values and finding the value of T2.

ΔT = 271.54/mol⋅K * (1/373 K - 1/T2)
ΔT = 271.54/mol⋅K * (T2 - 373 K) / (373 K * T2)

Now, we can substitute the pressure values:
ΔT = 271.54/mol⋅K * (T2 - 373 K) / (373 K * T2) = 1060 dyne/cm^2

Simplifying the equation:

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