Question

What is the heat change when \(1.25 \mathrm{~g}\) of water vapor (steam) at \(185.3^{\circ} \mathrm{C}\) is cooled to \(102.1^{\circ} \mathrm{C}^{\circ}\) ? The specific heat of steam is \(2.02 \mathrm{~J} /\left(\mathrm{g}{ }^{\circ} \mathrm{C}\right)\).

Short answer

The heat change when \(1.25 \mathrm{~g}\) of water vapor is cooled from \(185.3^{\circ} \mathrm{C}\) to \(102.1^{\circ} \mathrm{C}\) is \(210.9 \mathrm{~J}\).

Step-by-step solution

Identify the Variables

The mass 'm' of water vapor is \(1.25 \mathrm{~g}\), the specific heat 'c' is \(2.02 \mathrm{~J} /\left(\mathrm{g}{ }^{\circ} \mathrm{C}\right)\), and the change in temperature 'ΔT' is \(185.3^{\circ} \mathrm{C} - 102.1^{\circ} \mathrm{C}\).

Calculate ΔT - the Temperature Difference

Subtract the final temperature from the starting temperature: \(ΔT = 185.3^{\circ} \mathrm{C} - 102.1^{\circ} \mathrm{C} = 83.2^{\circ} \mathrm{C}\).

Calculate Heat Change

Now put the values into the formula: \(q = mcΔT = 1.25 \mathrm{~g} \times 2.02 \mathrm{~J} /\left(\mathrm{g}{ }^{\circ} \mathrm{C}\right) \times 83.2^{\circ} \mathrm{C}\).

Compute the Result

Multiplying the values from Step 3, we get \(q = 210.9 \mathrm{~J}\).

Key concepts

Specific Heat

The concept of specific heat is essential when understanding how substances change temperature. Specific heat refers to the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius.

This characteristic is unique to each material, determining how it responds to heat.In this exercise, the specific heat of steam is given as \(2.02 \mathrm{~J} / (\mathrm{g}^{\circ} \mathrm{C})\). This means that it takes \(2.02\) joules of energy to raise 1 gram of steam by 1 degree Celsius.

• Substances with higher specific heat require more energy to change their temperature.
• Water, for example, has a high specific heat, allowing it to absorb a lot of heat without a significant change in temperature.

Understanding specific heat helps in predicting and calculating heat changes in different materials.

Temperature Difference

Temperature difference, often abbreviated as ΔT, is the change in temperature experienced by a substance. It is crucial for warming or cooling processes, as it shows how much heat transfer has occurred.

In our given problem, we're cooling the steam from an initial \(185.3^\circ \mathrm{C}\) to a final \(102.1^\circ \mathrm{C}\).
This is calculated by subtracting the final temperature from the initial temperature: \[ ΔT = T_{\text{initial}} - T_{\text{final}} = 185.3^\circ \mathrm{C} - 102.1^\circ \mathrm{C} = 83.2^\circ \mathrm{C} \]

• Knowing ΔT helps in calculating other variables like heat change.
• It is a straightforward subtraction but is crucial for assessing energy transfer.

Understanding ΔT allows you to gauge how much a substance's thermal state has altered.

Heat Calculation

Calculating heat change involves using the relationship between mass, specific heat, and temperature difference. This fundamental concept is described by the formula:\[ q = mcΔT \]where:

• \(q\) is the heat change in joules.
• \(m\) represents mass in grams.
• \(c\) is the specific heat capacity in J/(g°C).
• \(ΔT\) is the temperature difference in Celsius.

For this exercise, substituting in the values gives:\[ q = 1.25\, \mathrm{g} \times 2.02\, \mathrm{J}/(\mathrm{g}^\circ \mathrm{C}) \times 83.2^\circ \mathrm{C} = 210.9 \mathrm{~J} \]Using this formula allows us to determine the energy absorbed or released during the temperature change. This calculation is critical in fields like thermodynamics and engineering, where understanding energy transfer is necessary for designing effective systems.