The molar heat capacity of a certain substance varies with...

The molar heat capacity of a certain substance varies with temperature according to the empirical equation $$C = 29.5 J/mol \cdot K + (8.20 \times 10{^-}{^3} J/mol \cdot K{^2})T$$ How much heat is necessary to change the temperature of 3.00 mol of this substance from 27$^\circ$C to 227$^\circ$C? (Hint: Use Eq. (17.18) in the form d$Q$ = n$C$ d$T$ and integrate.)

The molar heat capacity of a certain substance varies with temperature according to the empirical equation $$C = 29.5 J/mol \cdot K + (8.20 \times 10{^-}{^3} J/mol \cdot K{^2})T$$

How much heat is necessary to change the temperature of 3.00 mol of this substance from 27$^\circ$C to 227$^\circ$C? (Hint: Use Eq. (17.18) in the form d$Q$ = n$C$ d$T$ and integrate.)

Key Concepts

Heat Capacity

Heat capacity is a measure of the amount of heat required to change the temperature of a substance by a given amount. In thermodynamics, it is typically expressed per mole (molar heat capacity) when considering substances at the molecular level. This concept is fundamental in understanding how substances absorb or release heat during temperature changes.

Temperature Dependence of Heat Capacity

When heat capacity varies with temperature—as described by an empirical relation—it indicates that the energy required to effect a temperature change is not constant but changes with the temperature. This dependence must be taken into account by integrating the heat capacity function over the desired temperature range to accurately calculate the overall heat transfer.

Integration in Thermodynamics

In cases where variables such as heat capacity depend on temperature, integration becomes a key tool. The differential heat equation, dQ = n·C(T)dT, must be integrated over the specified temperature limits to compute the total amount of heat added or removed. This process consolidates the contributions of infinitesimal changes across a temperature range.

Temperature Conversion

Temperature conversion, typically from Celsius to Kelvin, is crucial since many thermodynamic equations are formulated in absolute temperature units (Kelvin). Proper conversion ensures the consistency and correctness of calculations, as the zero point in Kelvin is the absolute zero in thermodynamics.

Molar Quantity in Thermodynamic Calculations

The concept of molar quantities, such as using the number of moles in heat calculations, is central to thermodynamics. It allows the use of per-mole constants like molar heat capacity, thereby simplifying the calculation of heat required for temperature changes and providing a standardized measure that can be applied across different substances.

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