Question

You place \(100.0 \mathrm{g}\) of a hot metal in \(100.0 \mathrm{g}\) of cold water. Which substance (metal or water) undergoes a larger temperature change? Why is this?

Short answer

The hot metal undergoes a larger temperature change than the water when placed in contact with each other. This is because the metal has a lower specific heat capacity compared to water, meaning it takes less heat transfer to change the metal's temperature. The ratio of temperature changes is given by \(\frac{\Delta T_{metal}}{\Delta T_{water}} = \frac{c_{water}}{c_{metal}}\), with \(c_{water} > c_{metal}\), causing a larger temperature change in the metal.

Step-by-step solution

Understand Specific Heat Capacity and Heat Transfer

Specific heat capacity (c) is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 degree Celsius. Different substances have different specific heat capacities. For this exercise, we won't need the actual specific heat capacities, but just the comparison between them. When two substances with different temperatures come into contact, heat is transferred from the hotter substance to the colder substance until they reach thermal equilibrium. The total heat transferred (Q) can be calculated using the formula: \(Q = mc\Delta T\) where \(m\) is the mass of the substance, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature.

Set up the heat transfer equation

Since the heat lost by the hot metal is equal to the heat gained by the cold water, we can set up the equation with the heat transfer formula: \(m_{metal}c_{metal}\Delta T_{metal} = m_{water}c_{water}\Delta T_{water}\) Here, \(m_{metal}\) and \(m_{water}\) are both 100 grams, the masses of the metal and water, and \(c_{metal}\) and \(c_{water}\) represent the specific heat capacities of metal and water, respectively. \(\Delta T_{metal}\) and \(\Delta T_{water}\) are the temperature changes that we want to compare. Notice that the units of mass and specific heat capacities will cancel out when we compare the temperature changes.

Compare temperature changes using specific heat capacities

We will rearrange the equation for temperature change comparison: \(\frac{\Delta T_{metal}}{\Delta T_{water}} = \frac{m_{water}c_{water}}{m_{metal}c_{metal}}\) Since \(m_{metal} = m_{water}\), the masses will cancel out: \(\frac{\Delta T_{metal}}{\Delta T_{water}} = \frac{c_{water}}{c_{metal}}\) Now, we know that water has a higher specific heat capacity than most metals, meaning it takes more heat energy to raise the temperature of water than metal. Thus, in general, \(c_{water} > c_{metal}\).

Determine the substance with the larger temperature change

As we have found, the ratio of temperature changes depends only on specific heat capacities: \(\frac{c_{water}}{c_{metal}} > 1\) So, when we place the hot metal in the cold water, the metal will undergo a larger temperature change than the water, as it takes less heat transfer to change the metal's temperature compared to water. Therefore, the hot metal undergoes a larger temperature change due to its lower specific heat capacity compared to water.

Key concepts

Heat Transfer

Heat transfer is an essential concept in thermal physics, and it occurs when there is a temperature difference between two objects or substances. In the case of placing hot metal into cold water, heat energy flows from the warmer metal to the cooler water. This process continues until the temperatures of the metal and water equalize. There are three mechanisms of heat transfer: conduction, convection, and radiation. Here, conduction would be the primary mode, where the heat is transferred through direct contact between the metal and water.

Understanding the pathways and processes of heat transfer is crucial as it dictates how the energy moves from one substance to another, and it allows us to predict the outcome of thermal interactions. In everyday life, heat transfer is responsible for a range of phenomena, from the warming of our planet to the functioning of our home heating systems.

Temperature Change

Temperature change in a substance, often denoted as \(\Delta T\), is a measure of the difference between its initial and final temperatures. It is a direct indicator of the change in thermal energy of a substance. In our example, both the metal and the water experienced a temperature change due to heat transfer. However, the extent of this change depends on several factors, including the specific heat capacity of the substances involved.

The specific heat capacity refers to the amount of heat required to change the temperature of a unit mass of the substance by one degree. Substances with high specific heat capacities require more energy to change their temperatures than those with lower capacities. Consequently, water, with a high specific heat capacity, tends to undergo a smaller temperature change compared to metals when the same amount of heat is added or removed.

Thermal Equilibrium

Thermal equilibrium is achieved when two or more objects or substances in thermal contact no longer transfer heat, reaching a state where they all have the same temperature. When the hot metal is placed in cold water, the system strives to reach thermal equilibrium. The metal releases heat to the cooler water until the temperatures are equal. The nature of this equilibrium is dictated by the Zeroth Law of Thermodynamics, which states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

In the context of our exercise, achieving thermal equilibrium does not depend on equalizing the temperature changes of the metal and water but rather on the fact that the heat loss by the metal is equal to the heat gain by the water. This ensures that energy is conserved within the closed system of the two substances.