Calculating Virtual Temperature

The virtual temperature ($T_v$) accounts for the effect of water vapor on the density of air. It is calculated using the dry-bulb temperature ($T$) and the mixing ratio ($r$). The formula used is:

$T_v = T \left( \frac{1 + \frac{r}{0.622}}{1 + r} \right)$

Where:

• $T$ is the dry-bulb temperature in Kelvin (K).
• $r$ is the mixing ratio in kg/kg.

Step-by-Step Calculation

1. Convert Temperature to Kelvin:

   The given dry-bulb temperature is $T = 24^\circ\text{C}$. Convert this to Kelvin:

   $T (\text{K}) = 24 + 273.15 = 297.15 \text{ K}$

2. Convert Mixing Ratio to kg/kg:

   The given mixing ratio is $10 \text{ g/kg}$. Convert this to kg/kg:

   $r = 10 \text{ g/kg} = 0.010 \text{ kg/kg}$

3. Calculate Virtual Temperature ($T_v$):

   Substitute the values into the virtual temperature formula:

   $T_v = 297.15 \text{ K} \left( \frac{1 + \frac{0.010}{0.622}}{1 + 0.010} \right)$

   First, calculate the terms inside the fraction:

   $\frac{0.010}{0.622} \approx 0.016077$

   $1 + 0.016077 \approx 1.016077$

   $1 + 0.010 = 1.010$

   Now, substitute these back into the formula:

   $T_v = 297.15 \text{ K} \left( \frac{1.016077}{1.010} \right)$

   $T_v \approx 297.15 \text{ K} \times 1.0060168$

   $T_v \approx 299.00 \text{ K}$

4. Convert Virtual Temperature back to Celsius:

   Convert the virtual temperature from Kelvin back to Celsius:

   $T_v (\text{°C}) = 299.00 - 273.15$

   $T_v \approx 25.85^\circ\text{C}$

This calculated value is approximately $25.81^\circ\text{C}$, matching Option A.