Problem 15 - Chemistry Matter and Change

Question

Challenge If 335 g of water at \(65.5^{\circ} \mathrm{C}\) loses 9750 \(\mathrm{J}\) of heat, what is the final temperature of the water?

Step-by-Step Solution

Answer

The final temperature of the water is approximately \( 58.51^{\circ} \mathrm{C} \).

Step 1: Calculate the Temperature Change

To find the temperature change, use the formula: \[ q = mc\Delta T \]where \( q \) is the heat lost (9750 J), \( m \) is the mass of the water (335 g converted to 0.335 kg), \( c \) is the specific heat capacity of water (4.18 J/g°C, or 4180 J/kg°C), and \( \Delta T \) is the change in temperature. Rearrange the formula to solve for \( \Delta T \): \[ \Delta T = \frac{q}{mc} \]Substituting the values gives:\[ \Delta T = \frac{9750 \text{ J}}{0.335 \text{ kg} \times 4180 \text{ J/kg°C}} \approx 6.99 ^{\circ} \mathrm{C} \].

Step 2: Determine the Final Temperature

We know the initial temperature \( T_i = 65.5^{\circ} \mathrm{C} \) and the water loses heat, so the temperature decreases. The final temperature \( T_f \) can be found by subtracting the temperature change from the initial temperature:\[ T_f = T_i - \Delta T \]\[ T_f = 65.5^{\circ} \mathrm{C} - 6.99^{\circ} \mathrm{C} \approx 58.51^{\circ} \mathrm{C} \].

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics, which describes the flow of thermal energy from one object or material to another due to a difference in temperature. This process stops once thermal equilibrium is reached, meaning both systems reach the same temperature. There are three primary modes of heat transfer: conduction, convection, and radiation.

• **Conduction**: Transfer of heat through direct contact between molecules in solids.
• **Convection**: Transfer of heat through fluids (liquids or gases), where the fluid itself moves.
• **Radiation**: Transfer of heat through electromagnetic waves, without needing a medium (like the heat from the sun).

In our exercise, when water loses heat energy, it undergoes conduction as the energy is dispersed through its molecules. Understanding these modes helps in calculating how heat is exchanged, such as predicting how much heat a substance will lose or gain under certain conditions.

Temperature Change Calculation

Calculating temperature change involves understanding the relationship between heat energy, mass, and specific heat capacity. The equation used is: \[ q = mc\Delta T \]Where: - \( q \) is the heat energy (in Joules), - \( m \) is the mass of the substance (in kilograms), - \( c \) is the specific heat capacity (amount of heat per unit mass needed to raise the temperature by 1°C), - \( \Delta T \) is the temperature change (in Celsius). To find how much the temperature changes when heat is lost or gained, rearrange the formula to: \[ \Delta T = \frac{q}{mc} \]In the original problem, water’s temperature changes by \(6.99^{\circ} \mathrm{C}\) when it loses 9750 J of heat. Steps for calculation include ensuring units are compatible and carefully substituting values into the formula.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It involves studying how energy is converted between different forms and the laws governing these transformations.

• **First Law of Thermodynamics**: Often called the Law of Energy Conservation, it states that energy cannot be created or destroyed, only transformed.
• **Second Law of Thermodynamics**: Indicates that heat will naturally flow from hotter to cooler objects, and processes involving energy changes tend to progress toward thermal equilibrium.
• **Third Law of Thermodynamics**: As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.

In practical terms, these laws guide how we understand the cooling of water in our problem. When the water loses energy, it doesn't disappear. Instead, this energy is redistributed to the surroundings, illustrating the principle of energy conservation in thermodynamics. Furthermore, understanding these principles aids in accurately predicting and analyzing energy exchanges and their resulting effects.