A typical woman gives off heat at the rate of about 8000 calories per hour. How long would a woman have to stay in a bath of 60L (60,000g)of 26*C water in order to raise the water temperature to 30*C. Presume that all the heat given off by the woman is transferred to the water, and that the water doesn't lose any heat to the air
To calculate the time required for the woman to raise the temperature of the water in the bath, we need to use the principle of heat transfer and the specific heat capacity of water.
First, let's identify the known values:
- Initial water temperature (T1) = 26°C
- Final water temperature (T2) = 30°C
- Water mass (m) = 60000 grams
- Woman's heat release rate = 8000 calories per hour
We will follow these steps to find the answer:
Step 1: Convert the woman's heat release rate to an appropriate unit:
8000 calories per hour can be converted to Joules per second since 1 calorie = 4.184 Joules and 1 hour = 3600 seconds.
So, 8000 calories = 8000 * 4.184 Joules = 33472 Joules per hour = 33472/3600 Joules per second, which is approximately 9.297 Joules per second.
Step 2: Calculate the total heat transfer required to raise the water temperature:
The heat transfer required can be calculated using the formula:
Q = m * c * ΔT,
where Q is the heat transfer, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.
For water, the specific heat capacity (c) is approximately 4.184 Joules per gram per °C.
So, ΔT = T2 - T1 = 30°C - 26°C = 4°C.
Plugging in the values, we get:
Q = 60000 grams * 4.184 J/g°C * 4°C = 1001760 Joules.
Step 3: Determine the time required:
To find the time required, we use the formula:
Q = P * t,
where Q is the heat transfer, P is the power, and t is the time.
In this case, the heat transfer is 1001760 Joules, and the power (P) is 9.297 Joules per second.
Substituting the values, we get:
1001760 = 9.297 * t.
Now, solve for t:
t = 1001760 / 9.297 ≈ 107787 seconds.
Therefore, a woman would need to stay in the bath for approximately 107787 seconds or about 29.94 hours to raise the water temperature from 26°C to 30°C, assuming all the heat given off by the woman is transferred to the water and no heat is lost to the air.