Question

A girl has discovered a very old U.S. silver dollar and is holding it tightly in her little hands. Suppose that she put the silver dollar on the wooden (insulating) surface of a table, and then a friend came in from outside and placed on top of the silver dollar an equally old penny that she just found in the snow, where it had been left all night. Estimate the final equilibrium temperature of the system of the two coins in thermal contact.

Short answer

Based on the provided solution, here is a potential short answer: The final equilibrium temperature of the two coins (a silver dollar and a penny) in thermal contact is approximately 17.3°C. This was determined by applying the conservation of energy principle. Given the initial temperatures and the specific heat capacities of silver and copper, the heat gained by the silver dollar was equated to the heat lost by the penny. By solving the resulting equation, we found the final equilibrium temperature.

Step-by-step solution

Identify the given information and assume initial temperatures

In the problem, we know that: 1. The silver dollar is at room temperature (let's assume 20°C). 2. The penny's initial temperature is equal to the snow temperature (let's assume 0°C). Let T be the final equilibrium temperature.

Find specific heat capacities of silver and copper

The specific heat capacity of silver \(c_{silver}\) is approximately 235 J/(kg·K) and the specific heat capacity of copper \(c_{copper}\) (copper is the main component of a penny) is approximately 390 J/(kg·K).

Calculate the masses of the coins

The mass of a silver dollar \(m_{silver}\) is approximately 26.73 g, and the mass of a penny \(m_{copper}\) is approximately 2.5 g. Convert these to kg, have: \(m_{silver} = 0.02673\) kg \(m_{copper} = 0.0025\) kg

Apply conservation of energy and solve for T

According to the conservation of energy principle, the heat gained by the silver dollar will be equal to the heat lost by the penny. We can write this as: \(m_{silver}c_{silver}(T - T_{initial, silver}) = m_{copper}c_{copper}(T_{initial, copper} - T)\) Now, plug in the values of \(m_{silver}\), \(c_{silver}\), \(T_{initial, silver}\), \(m_{copper}\), \(c_{copper}\), and \(T_{initial, copper}\). Then solve for T. \(0.02673(235)(T - 20) = 0.0025(390)(0 - T)\) Simplifying the equation and solve for T: \(6.27155T - 125.431 = -975T\) Adding 125.431 to both sides and combining the T terms: \(7.24655T = 125.431\) Now, divide by 7.24655 to solve for T: \(T = \frac{125.431}{7.24655} = 17.3 °C\)

Final answer

The final equilibrium temperature of the two coins in thermal contact is approximately 17.3°C.