Calculate the change in entropy of 1.00 kg of water when it is heated from 0°C to 85°C. Use the integral ∆S = ∫ dQ / T.

Verified step by step guidance

1. Step 1: Recall the formula for the change in entropy, which is given by \( \Delta S = \int \frac{dQ}{T} \). Here, \( dQ \) is the infinitesimal amount of heat added, and \( T \) is the absolute temperature in kelvins.
2. Step 2: Express \( dQ \) in terms of the specific heat capacity \( c \), mass \( m \), and temperature change \( dT \). The relationship is \( dQ = mc \, dT \). For water, \( c = 4186 \, \text{J/(kg·K)} \) and \( m = 1.00 \, \text{kg} \).
3. Step 3: Substitute \( dQ = mc \, dT \) into the entropy formula: \( \Delta S = \int \frac{mc \, dT}{T} \). Since \( m \) and \( c \) are constants, they can be factored out: \( \Delta S = mc \int \frac{dT}{T} \).
4. Step 4: Evaluate the integral \( \int \frac{dT}{T} \), which is the natural logarithm: \( \int \frac{dT}{T} = \ln \left( \frac{T_2}{T_1} \right) \). Here, \( T_1 \) is the initial temperature (0°C = 273 K) and \( T_2 \) is the final temperature (85°C = 358 K).
5. Step 5: Combine the results to find \( \Delta S \): \( \Delta S = mc \ln \left( \frac{T_2}{T_1} \right) \). Substitute the values for \( m \), \( c \), \( T_1 \), and \( T_2 \) to calculate the change in entropy. Ensure the temperatures are in kelvins.

Key Concepts

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the amount of energy in a physical system that is not available to do work. The change in entropy (∆S) reflects how energy is distributed among the particles in a system, particularly during processes like heating or phase changes.

Heat Transfer (dQ)

Heat transfer, denoted as dQ, refers to the amount of thermal energy exchanged between a system and its surroundings. In the context of heating water, dQ represents the energy added to the water as it is heated from one temperature to another. This energy change is crucial for calculating the change in entropy, as it directly influences the system's thermal state.

Temperature (T)

Temperature (T) is a measure of the average kinetic energy of the particles in a substance. In the context of the entropy formula ∆S = ∫ dQ / T, temperature plays a critical role as it affects how much entropy change occurs for a given amount of heat transfer. As temperature increases, the impact of added heat on entropy diminishes, illustrating the relationship between heat, temperature, and disorder.