Problem Statement
Air at $T=25°\text{C}$ has a dew point of $T_d=15°\text{C}$. The saturation vapour pressures are $p_s(25°)=3167\ \text{Pa}$ and $p_s(15°)=1705\ \text{Pa}$. Find the relative humidity.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The dew point is the temperature at which the actual vapour pressure equals the saturation pressure. Thus the actual vapour pressure at 25°C is:
Step 2 — Apply the relevant physical law or equation: $$p_{vapour} = p_s(15°C) = 1705\ \text{Pa}$$
Step 3 — Solve algebraically for the unknown: Relative humidity:
Step 4 — Substitute numerical values with units: $$RH = \frac{p_{vapour}}{p_s(T)} = \frac{1705}{3167} \approx 0.538 = 53.8\%$$
Step 5 — Compute and check the result: Result: Relative humidity $\approx 54\%$.
Worked Calculation
$$p_{vapour} = p_s(15°C) = 1705\ \text{Pa}$$
$$RH = \frac{p_{vapour}}{p_s(T)} = \frac{1705}{3167} \approx 0.538 = 53.8\%$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
The dew point is the temperature at which the actual vapour pressure equals the saturation pressure. Thus the actual vapour pressure at 25°C is:
$$p_{vapour} = p_s(15°C) = 1705\ \text{Pa}$$
Relative humidity:
$$RH = \frac{p_{vapour}}{p_s(T)} = \frac{1705}{3167} \approx 0.538 = 53.8\%$$
Result: Relative humidity $\approx 54\%$.
Answer
$$\boxed{RH = \frac{p_{vapour}}{p_s(T)} = \frac{1705}{3167} \approx 0.538 = 53.8\%}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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