Question

A copper sphere with density 8900 kg/m3, radius 5. 00 cm, and emissivity e = 1. 00 sits on an insulated stand. The initial temperature of the sphere is 300 K. The surroundings are very cold, so the rate of absorption of heat by the sphere can be neglected.

a. How long does it take the sphere to cool by 1. 00 K due to its radiation of heat energy? Neglect the change in heat current as the temperature decreases.

b. To assess the accuracy of the approximation used in part A, what is the fractional change in the heat current H when the temperature changes from 300 K to 299 K?

Answer 1

To calculate the cooling time of the copper sphere and assess the accuracy of the approximation, we can use equations and given values to determine the necessary variables.

Step-by-step explanation:

a) To calculate how long it takes for the copper sphere to cool by 1.00 K due to radiation, we can use Newton's law of cooling:

ΔT = -k * Δt

Where ΔT is the change in temperature, Δt is the change in time, and k is the cooling constant. The cooling constant can be calculated using the equation:

k = (e * σ * A * ΔT⁴) / (m * C)

Where e is the emissivity, σ is the Stefan-Boltzmann constant, A is the surface area of the sphere, ΔT is the change in temperature, m is the mass of the sphere, and C is the specific heat capacity of copper.

By substituting the given values into the equation, we can solve for Δt.

b) To assess the accuracy of the approximation used in part a, we can calculate the fractional change in the heat current H using the equation:

ΔH / H = (ΔT / T)

Where ΔH is the change in heat current, H is the initial heat current, ΔT is the change in temperature, and T is the initial temperature. By substituting the given values into the equation, we can calculate the fractional change in the heat current when the temperature changes from 300 K to 299 K.