Question

A \(45-\mathrm{kg}\) sample of water absorbs \(345 \mathrm{~kJ}\) of heat. If the water was initially at \(22.1{ }^{\circ} \mathrm{C}\), what is its final temperature?

Short answer

The final temperature of the water is approximately 30.3°C.

Step-by-step solution

Understand the Concept of Specific Heat Capacity

The specific heat capacity of a substance is the amount of heat required to raise the temperature of 1 kilogram of the substance by 1 degree Celsius. Water has a specific heat capacity of approximately 4.18 kJ/kg°C.

Calculate the Temperature Change

Use the formula for heat transfer, Q = mcΔT, where Q is the amount of heat absorbed, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature. Rearrange the formula to solve for ΔT: ΔT = Q / (mc).

Solve for the Final Temperature

To find the final temperature (T_final), add the change in temperature (ΔT) to the initial temperature (T_initial): T_final = T_initial + ΔT.

Key concepts

Heat Transfer

When we talk about heat transfer, we're referring to the process by which thermal energy moves from one object or material to another. This process can occur in various ways, including conduction, convection, and radiation. Conduction happens through direct contact, convection through the movement of fluids such as air or water, and radiation through electromagnetic waves.

In our exercise, the heat transfer takes place when the water absorbs thermal energy. What's interesting about water is that it has a relatively high specific heat capacity, which means it can absorb a lot of heat before it begins to warm up significantly. This characteristic is crucial in many applications, including climate regulation and cooking.

Temperature Change Calculation

Understanding how to calculate a temperature change is essential in many fields, from engineering to meteorology. It's basically figuring out how much an object's temperature will change when it gains or loses heat.

Here's a simple breakdown:

• Measure the amount of heat absorbed (Q), typically in joules or kilojoules.
• Determine the mass (m) of the substance, often in kilograms.
• Identify the specific heat capacity (c) of the substance, the energy needed to change the temperature of 1 kg of the substance by 1°C.
• Use the formula \( \Delta T = \frac{Q}{mc} \) to find the change in temperature (ΔT).

In the exercise, we used these steps to find out that a 45-kg sample of water absorbed 345 kJ of heat. By inserting the specific heat capacity of water and rearranging the formula, we calculated the rise in temperature, which helped us find the water's final temperature.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationship between heat and other forms of energy. In essence, it's about understanding how thermal energy is converted to and from other energy types, and how it affects matter.

Our exercise is a classic example of the first law of thermodynamics, which is the principle of energy conservation. The energy doesn't disappear; instead, it transforms from one form to another—in this case, from stored chemical energy in a hot plate, perhaps, to thermal energy in water.

This energetic transformation directly influences the temperature of the water as we've established with our temperature change calculation. These principles form the foundation of countless processes, from industrial operations to the biological functions that keep us alive.