(a) How much heat is required to raise the temperature of...

(a) How much heat is required to raise the temperature of 1000 grams of nitrogen from $-20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ at constant pressure? (b) How much has the internal energy of the nitrogen increased? (c) How much external work was done? (d) How much heat is required if the volume is kept constant? Take the specific heat at constant volume $c_v=5 \mathrm{cal} / \mathrm{mole}^{\circ} \mathrm{C}$ and $R=2 \mathrm{cal} / \mathrm{mole} \cdot{ }^{\circ} \mathrm{C}$.

(a) How much heat is required to raise the temperature of 1000 grams of nitrogen from $-20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ at constant pressure?
(b) How much has the internal energy of the nitrogen increased?
(c) How much external work was done?
(d) How much heat is required if the volume is kept constant?

Take the specific heat at constant volume $c_v=5 \mathrm{cal} / \mathrm{mole}^{\circ} \mathrm{C}$ and $R=2 \mathrm{cal} / \mathrm{mole} \cdot{ }^{\circ} \mathrm{C}$.

Key Concepts

Specific Heat Capacities

This concept involves the amount of heat required to raise the temperature of a substance by one degree. In thermodynamics, there are two main types: the specific heat at constant volume (c_v) and the specific heat at constant pressure (c_p). For ideal gases, these two are related by the equation c_p = c_v + R, where R is the gas constant. Understanding these values is crucial for calculating the energy changes in processes that occur at either constant volume or constant pressure.

First Law of Thermodynamics

The first law of thermodynamics states that energy cannot be created or destroyed, only transferred or converted from one form to another. It is typically expressed as ?U = Q - W, where ?U is the change in internal energy of the system, Q is the heat added to the system, and W is the work done by the system. This principle is key for analyzing energy balance in thermodynamic processes by determining the contributions of heat and work to changes in internal energy.

Work in Thermodynamic Processes

In thermodynamics, work is done by the system when it expands or contracts against an external pressure. At constant pressure, the work done by an ideal gas can be expressed as W = P?V, where P is the pressure and ?V is the change in volume. The concept of work is important for relating the energy inputs (heat) to the observable mechanical effects in a system undergoing expansion or compression.

Ideal Gas Law

The ideal gas law, given by PV = nRT, relates the pressure, volume, and temperature of an ideal gas to the number of moles (n) present and the gas constant (R). This law is essential for calculating how changes in one variable affect the others during thermodynamic processes, such as heating at constant pressure or constant volume. It provides a framework for connecting macroscopic measurements to microscopic behavior in gases.

Calculation of Moles from Mass

When working with substances in thermodynamic problems, it is often necessary to convert mass to number of moles since many properties, like specific heat capacities, are given per mole. This conversion is done using the molar mass of the substance. Knowing the number of moles is crucial for applying equations like the ideal gas law and for calculating the total heat required or the change in internal energy in a system.

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