A person sitting quietly in a room on a winter day may lose heat primarily by radiation from the outer surface of her clothes to the walls of the room. If the walls are at 13.8° C and the rate of radiation is 96.0 kcal/h (a typical basal metabolic rate), what is the temperature of the outer surface of the clothes, which have an emissivity of 0.800? (Assume the surface area of the person is 1 m2.)

To solve this problem, we can use the Stefan-Boltzmann law, which relates the rate of radiation emitted by an object to its temperature and emissivity.

The Stefan-Boltzmann law states that the rate of energy radiated by an object is proportional to the fourth power of its temperature multiplied by its emissivity. Mathematically, it can be expressed as:

P = εσA(T^4 - T0^4)

Where:
- P is the rate of radiation (given as 96.0 kcal/h)
- ε is the emissivity of the clothes (given as 0.800)
- σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/(m^2K^4))
- A is the surface area of the person (approximately 1 m^2)
- T is the temperature of the outer surface of the clothes (unknown variable)
- T0 is the temperature of the walls (given as 13.8°C which is equal to 13.8 + 273.15 = 286.95 K)

Let's plug in the given values and solve for T:

96.0 kcal/h = 0.800 (5.67 x 10^-8 W/(m^2K^4)) (1 m^2) (T^4 - (286.95 K)^4)

First, convert kcal/h to watts (W):
1 kcal/h = 1 x 4.2 kJ/h = 4.2 x 10^3 J/h = 4.2 x 10^3 / 3600 J/s = 1.1667 W

Now we can solve for T:

1.1667 W = 0.800 (5.67 x 10^-8 W/(m^2K^4)) (1 m^2) (T^4 - (286.95 K)^4)

Simplifying, we have:

1.1667 = 0.800 (5.67 x 10^-8) (T^4 - (286.95)^4)

Dividing both sides by 0.800 (5.67 x 10^-8), we get:

1.1667 / (0.800 (5.67 x 10^-8)) = T^4 - (286.95)^4

Solving for T^4, we have:

T^4 = (1.1667 / (0.800 (5.67 x 10^-8))) + (286.95)^4

Taking the fourth root of both sides, we get:

T = ((1.1667 / (0.800 (5.67 x 10^-8))) + (286.95)^4)^(1/4)

Calculating this expression will give us the value for T, which represents the temperature of the outer surface of the clothes.