Question

A 2.8-kg sample of a metal with a specific heat of 0.127 cal/g℃ is heated to 100℃ then placed in a 50 grams sample of water at 30 ℃. What is the final temperature of the metal and the water?
Solution

Answer 1

First, let's determine the amount of heat gained by water (Q_water).
We will use the formula for heat transfer: Q = mcΔT, where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

1. Given values for water: m_water = 50 g, c_water = 1 cal/g℃, and initial temperature T1_water = 30℃
2. We will assume the final temperature to be Tf (which we need to calculate). So, ΔT_water = Tf - T1_water

Now, let's find the amount of heat lost by the metal (Q_metal).
1. Given values for the metal: m_metal = 2.8 kg = 2800 g, c_metal = 0.127 cal/g℃, and initial temperature T1_metal = 100℃
2. The final temperature for the metal will also be Tf. So, ΔT_metal = Tf - T1_metal

Since the heat lost by the metal is equal to the heat gained by water, we can equate both Q_metal and Q_water:
Q_metal = Q_water

By substituting the formula for heat transfer, we get:
m_metal * c_metal * ΔT_metal = m_water * c_water * ΔT_water

Now, substitute the given values and simplify the equation:
2800 * 0.127 * (Tf - 100) = 50 * 1 * (Tf - 30)

357.6 * (Tf - 100) = 50 * (Tf - 30)

Now, expand the equation:
357.6 * Tf - 35760 = 50 * Tf - 1500

Subtract 50 * Tf from both sides:
307.6 * Tf - 35760 = -1500

Now, add 35760 to both sides:
307.6 * Tf = 34260

Finally, divide by 307.6:
Tf = 34260 / 307.6
Tf ≈ 111.36℃

However, we need to consider that the metal's temperature will decrease, and the water's temperature will increase. Therefore, the final temperature should be between the initial temperatures of both substances (30℃ and 100℃).

Since our calculated temperature (111.36℃) is outside this range, there must be an error in our calculations or given values.

It's important to note that the specific heat of the metal (0.127 cal/g℃) is much lower than water's (1 cal/g℃). This means that the metal will lose heat relatively quickly compared to the water, making it unlikely that the water would heat up above the metal's initial temperature (100℃).

Therefore, the final temperature of the metal and the water cannot be accurately determined with the given information. There might be some error in the given values or the problem statement.