When 20.9 J of thermal energy was added to a particular...

When $20.9 \mathrm{~J}$ of thermal energy was added to a particular ideal gas, the volume of the gas changed from $50.0$ $\mathrm{cm}^{3}$ to $100 \mathrm{~cm}^{3}$ while the pressure remained constant at $1.00 \mathrm{~atm}$. (a) By how much did the internal energy of the gas change? If the quantity of gas present is $2.00 \times 10^{-3} \mathrm{~mol}$, find the molar specific heat of the gas at (b) constant pressure and (c) constant volume.

When $20.9 \mathrm{~J}$ of thermal energy was added to a particular ideal gas, the volume of the gas changed from $50.0$ $\mathrm{cm}^{3}$ to $100 \mathrm{~cm}^{3}$ while the pressure remained constant at $1.00 \mathrm{~atm}$.
(a) By how much did the internal energy of the gas change? If the quantity of gas present is $2.00 \times 10^{-3} \mathrm{~mol}$, find the molar specific heat of the gas at (b) constant pressure and (c) constant volume.

Key Concepts

First Law of Thermodynamics

The First Law of Thermodynamics states that energy is conserved in any process, meaning that the change in a system's internal energy is equal to the heat added to the system minus the work done by the system on its surroundings. This principle is fundamental in thermodynamics and is applied to many processes, such as the heating of an ideal gas, where it helps in relating the transferred heat to the changes in internal energy and work done during expansion or compression.

Work Done during Constant Pressure Expansion

When an ideal gas expands at constant pressure, the work done by the gas is calculated as the product of the constant pressure and the change in volume (W = P?V). This concept is crucial because it quantifies the energy spent by the gas in doing work on its environment, which directly influences the internal energy change of the gas as described by the First Law of Thermodynamics.

Molar Specific Heat

Molar specific heat is the amount of heat required to raise the temperature of one mole of a substance by one degree in a given process. It is an important property in thermodynamics that helps quantify how substances absorb heat under different conditions, such as at constant pressure (C?) or constant volume (C?). This property is used to understand and predict the thermal behavior of gases and other materials during heating or cooling.

Relationship between C? and C? in Ideal Gases

For ideal gases, there is a well-known relationship between the specific heats at constant pressure and constant volume given by C? - C? = R, where R is the universal gas constant. This relation is derived from the underlying principles of energy distribution and molecular motion, and it allows for the determination of one specific heat if the other is known, thereby connecting thermal properties with the molecular characteristics of the gas.

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