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 raise the temperature of that much water by 4 degrees C requires
Q = 4 deg* 1.00 cal/deg*g * 6*10^4 g
= 24*10^4 calories
Divide that by 8000 cal/h for the number of hours.
The bath water will probably lose heat much faster than that, by evaporation and convection. This is an unrealistic problem.
To calculate the time it would take for a woman to raise the temperature of the bathwater, we need to consider the amount of heat transferred from the woman to the water.
First, let's determine the heat required to raise the temperature of the water from 26°C to 30°C. This can be done using the specific heat capacity formula:
Q = m * c * ΔT
Where:
Q = Heat energy transferred
m = Mass of the water
c = Specific heat capacity of water
ΔT = Change in temperature
Given:
m = 60,000g = 60kg (since 1L of water has a mass of 1kg)
c = 4.18 J/g°C (specific heat capacity of water)
ΔT = 30°C - 26°C = 4°C
Plugging in the values:
Q = 60kg * 4.18 J/g°C * 4°C
Q = 1003,2 J
Now, let's calculate the time using the rate at which the woman gives off heat. Given that a typical woman gives off heat at a rate of 8000 calories per hour, we need to convert this to Joules:
1 calorie = 4.18 Joules
8000 calories/hour = 8000 * 4.18 J/hour
To find out how long the woman would need to stay in the bath, we'll divide the heat required to raise the bathwater's temperature by the rate at which the woman gives off heat:
Time = Heat Required / Heat Rate
Time = 1003.2 J / (8000 * 4.18 J/hour)
Simplifying the equation:
Time = 1003.2 / (33440 J/hour)
Time ≈ 0.03 hours
Converting to minutes:
Time ≈ 0.03 hours * 60 minutes/hour
The woman would need to stay in the bath for approximately 1.8 minutes to raise the water temperature from 26°C to 30°C, assuming all the heat given off is transferred to the water and no heat is lost to the air.