A gas in a cylinder was placed in a heater and gained \(7000 \mathrm{~kJ}\) of heat. If the cylinder increased in volume from \(700 \mathrm{~mL}\) to \(1450 \mathrm{~mL}\) against an atmospheric pressure of 750 Torr during this process, what is the change in internal energy of the gas in the cylinder?

The first law of thermodynamics, often referred to as the law of energy conservation, is a fundamental principle that describes the relationship between the heat added to a system and the work done by the system. It asserts that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the work done by the system on its surroundings.

This law can be mathematically represented as \( \Delta U = Q - W \), where \( \Delta U \) stands for the change in internal energy, \( Q \) for the heat added to the system, and \( W \) for the work done by the system. It is a pivotal concept in physics and chemistry that helps us understand how different forms of energy are interconvertible and the energy balance within a system.

When we talk about 'work done by gas' in thermodynamics, we refer to the energy transferred when a gas expands or contracts within a container. To calculate the work done by the gas during expansion or compression, the following equation is often used: \( W = -P \Delta V \), where \( W \) is the work done by the gas, \( P \) is the pressure, and \( \Delta V \) is the change in volume of the gas. It is important to note that work is considered positive when done on the system and negative when done by the system.

This concept is significant because it connects mechanical work with thermal processes. Understanding how to calculate work is critical for predicting how systems will behave when energy is added or removed. In the context of an expanding gas, as in our exercise, work done by the system on its surroundings occurs as the gas pushes against an external pressure to increase its volume.

Heat transfer is a process by which energy, due to a temperature difference, is exchanged either within a system or between different systems. There are three modes of heat transfer: conduction, convection, and radiation. In the context of our exercise, we are concerned with the amount of heat acquired by the gas when placed in a heater.

The measure of this heat, usually in joules or kilojoules, can directly influence the system's temperature and phase. It's essential when calculating the change in internal energy of a system to know precisely how much heat is added or removed. In the given problem, the gas gains \( 7000 \mathrm{~kJ} \) of heat from the heater, which is a significant factor in determining the resulting change in internal energy.

Gas expansion occurs when a gas's volume increases. The gas molecules move faster and spread out, pushing against the confines of their container. This behavior is inherently related to temperature and pressure. When a gas heats up, the increased kinetic energy of the particles causes them to push outwards, leading to expansion.

In our exercise, the expansion is mechanically resisted by the atmospheric pressure, evidenced by the opposing force as the gas does work against this pressure. The difference in volume before and after the heating indicates the degree of expansion. Mathematically, the change in volume \( (\Delta V) \) is a crucial variable both in calculating work done by the gas and understanding the physical changes occurring within the system. The expansion of a gas is not only a fundamental concept in thermodynamics but also a critical element of many engineering and scientific applications, such as in engines and refrigeration systems.