Thermal Energy Transfer and Specific Heat Capacity

Fundamentals of Specific Heat Capacity

• Specific Heat Capacity Definition: This is formally defined as the amount of heat energy required to raise the temperature of $1\,g$ of a specific substance by $1^{\circ}C$.

• Inverse Relationship with Heating Time: There is a direct correlation between the specific heat capacity of a material and the time it takes to change its temperature:

  • The higher the specific heat capacity of a substance, the longer it takes to heat that substance.

  • Conversely, a lower specific heat capacity means the substance will heat up (or cool down) much more quickly.

• Properties of Water: It is vital to note that water possesses a very high specific heat capacity compared to many other common substances.

The Specific Heat Capacity Formula

• Usage Constraints: This formula is utilized exclusively for calculating energy changes during temperature changes (sensible heat) and does not apply to phase changes.

• Mathematical Representation:     $Q = mc\Delta T$

• Variable Breakdown and Units:

  • $m$: Refers to the mass of the substance. This must be measured in grams ($g$). If the initial value is provided in kilograms ($kg$), you must convert it to grams first ($1\,kg = 1000\,g$).

  • $c$: Refers to the specific heat capacity of the substance. The units are Joules per gram-degree Celsius ($J/g^{\circ}C$). These values are typically sourced from a standard data booklet.

  • $\Delta T$: Refers to the change in temperature, measured in degrees Celsius (${\circ}C$). This is calculated as Final Temperature minus Initial Temperature ($T_{final} - T_{initial}$).

  • $Q$: Refers to the total amount of heat or thermal energy transferred, measured in Joules ($J$).

• Sign Conventions for Energy Transfer ($Q$):

  • When $Q$ is positive ($+ve$), it indicates that heat or energy is being absorbed by the system/substance.

  • When $Q$ is negative ($-ve$), it indicates that heat or energy is being released by the system/substance.

Specific Heat Capacity Values for Various Substances

• The following values reflect the specific heat capacities ($c$) in units of $J/g^{\circ}C$ for substances found in nature:

  • Pure water: $4.19\,J/g^{\circ}C$

  • Sea water: $3.89\,J/g^{\circ}C$

  • Steam: $2.02\,J/g^{\circ}C$

  • Ice: $2.00\,J/g^{\circ}C$

  • "Moist" air: $1.15\,J/g^{\circ}C$ (Note: this value varies depending on humidity levels).

  • "Dry" air: $1.00\,J/g^{\circ}C$

Computational Examples

• Example 1: Heating Dry Air in a Household

  • Scenario: A house contains $170\,kg$ of dry air. The furnace has broken, causing the air temperature to fall to $2.0^{\circ}C$. How much energy is needed to heat the air back up to $20^{\circ}C$?

  • Identification of Givens:

    • Mass ($m$) = $170\,kg = 170,000\,kg$ (Wait, correct conversion: $170\,kg \times 1000 = 170,000\,g$).

    • Specific Heat of Dry Air ($c$) = $1.00\,J/g^{\circ}C$.

    • Initial Temp ($T_i$) = $2.0^{\circ}C$.

    • Final Temp ($T_f$) = $20^{\circ}C$.

    • Temperature Change ($\Delta T$) = $20^{\circ}C - 2.0^{\circ}C = 18^{\circ}C$.

  • Calculation:         $Q = (170,000\,g) \times (1.00\,J/g^{\circ}C) \times (18^{\circ}C) = 3,060,000\,J$ or $3.06 \times 10^{6}\,J$.

• Example 2: Cooling of Glass

  • Scenario: Suppose that there is $10.7\,g$ of glass initially at $7.0^{\circ}C$. What would be the final temperature of the glass if it released $150\,J$ of energy when heated?

  • Identification of Givens:

    • Mass ($m$) = $10.7\,g$.

    • Initial Temp ($T_i$) = $7.0^{\circ}C$.

    • Energy Released ($Q$) = $-150\,J$ (Note: Even though the transcript says "released… when heated," a release of energy implies a negative $Q$ value for the substance in thermal calculations).

    • Specific Heat of Glass ($c$): To be retrieved from a data booklet.

Assignments and Further Practice

• Homework Task: Complete the following exercises from the textbook to reinforce understanding of thermal energy transfer:

  • Reference: Page 377.

  • Questions: Problems 1 through 9.