To solve the problem of how long it will take to raise the temperature of the air in the classroom from 20°C to 37°C with 50 people losing energy at an average of 150 watts, we can follow these steps:
### Step 1: Calculate the volume of the room
The dimensions of the classroom are given as 5 m x 10 m x 3 m.
\[
\text{Volume} = 5 \, \text{m} \times 10 \, \text{m} \times 3 \, \text{m} = 150 \, \text{m}^3
\]
### Step 2: Convert the volume to liters
Since 1 m³ = 1000 liters, we convert the volume:
\[
\text{Volume in liters} = 150 \, \text{m}^3 \times 1000 \, \text{L/m}^3 = 150000 \, \text{L}
\]
### Step 3: Calculate the number of moles of air in the room
Using the ideal gas equation \(PV = nRT\), we can find the number of moles (n). Given:
- Pressure (P) = 1 atm = 101.325 kPa
- Volume (V) = 150 m³ = 150,000 L
- R (ideal gas constant) = 0.0821 L·atm/(K·mol)
- Temperature (T) = 20°C = 293 K
Convert pressure to atm for consistency:
\[
n = \frac{PV}{RT} = \frac{(1 \, \text{atm}) \times (150000 \, \text{L})}{(0.0821 \, \text{L·atm/(K·mol)}) \times (293 \, \text{K})}
\]
Calculating this gives:
\[
n \approx 6.236 \times 10^3 \, \text{mol}
\]
### Step 4: Calculate the heat required to raise the temperature
We need to calculate the heat required to raise the temperature from 20°C to 37°C.
\[
\Delta T = T_2 - T_1 = 37°C - 20°C = 17°C
\]
Given \(C_P = \frac{7}{2}R\), we can calculate \(C_P\):
\[
C_P = \frac{7}{2} \times 8.314 \, \text{J/(mol·K)} \approx 29.1 \, \text{J/(mol·K)}
\]
Now, the total heat (Q) required is:
\[
Q = n \cdot C_P \cdot \Delta T
\]
\[
Q = (6.236 \times 10^3 \, \text{mol}) \cdot (29.1 \, \text{J/(mol·K)}) \cdot (17 \, \text{K})
\]
\[
Q \approx 3.085 \times 10^6 \, \text{J}
\]
### Step 5: Calculate the total power output from the people
Each person loses energy at an average of 150 watts. For 50 people:
\[
\text{Total Power} = 50 \, \text{people} \times 150 \, \text{W} = 7500 \, \text{W}
\]
### Step 6: Calculate the time required to raise the temperature
Using the formula for time:
\[
t = \frac{Q}{\text{Power}} = \frac{3.085 \times 10^6 \, \text{J}}{7500 \, \text{W}} \approx 411.3 \, \text{s}
\]
### Final Answer
The time needed to raise the temperature of the air in the room to body temperature is approximately **411.3 seconds**.
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