A balloon filled with 39.1 mol helium has a volume of 876 L at 0.0^∘ C and 1.00 atm pressure. T

step 1 First of all, to calculate heat q, so that will be given by that is a relation n, multiplied by

A balloon filled with 39.1 mol helium has a volume of 876 L...

A balloon filled with $39.1$ mol helium has a volume of $876 \mathrm{~L}$ at $0.0^{\circ} \mathrm{C}$ and $1.00$ atm pressure. The temperature of the balloon is increased to $38.0^{\circ} \mathrm{C}$ as it expands to a volume of $998 \mathrm{~L}$, the pressure remaining constant. Calculate $q, w$, and $\Delta E$ for the helium in the balloon. (The molar heat capacity for helium gas is $20.8 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot$ mol. $)$

Key Concepts

First Law of Thermodynamics

The First Law of Thermodynamics states that the change in a system's internal energy is equal to the heat added to the system plus the work done on the system. This principle, symbolized as ?E = q + w, is a foundational concept in energy conservation and helps relate heat transfer, work, and changes in internal energy during any thermodynamic process.

Work Done at Constant Pressure

When a gas expands or contracts at constant pressure, the work done by the gas is calculated using the formula w = -P?V, where P is the external pressure and ?V is the change in volume. The negative sign indicates that work done by the system (expansion) is considered as energy leaving the system, which is an important convention in thermodynamics.

Heat Capacity of a Gas

Heat capacity describes the amount of heat required to change the temperature of a substance by a given amount. For gases, the molar heat capacity at constant pressure (Cp) is used when the process involves heating at constant pressure. It quantifies the relationship between heat added (q) and the temperature change (?T), allowing one to calculate q = nCp?T for a given number of moles n.

Internal Energy Change for an Ideal Gas

For an ideal gas, the change in internal energy is directly related to the change in temperature and is calculated using the molar heat capacity at constant volume (Cv), such that ?E = nCv?T. Since Cp and Cv are related through the gas constant (Cp = Cv + R), knowing one and the temperature change allows for determining the internal energy change during a process.

Ideal Gas Behavior

Ideal gas behavior assumes that the gas molecules do not interact and that the volume occupied by the molecules themselves is negligible compared to the container volume. This assumption simplifies relationships between pressure, volume, temperature, and the amount of gas, making it easier to apply formulas such as the ideal gas law and to compute quantities like work, heat, and internal energy in thermodynamic processes.

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