Question

What is the change in internal energy (in \(J\) ) of a system that absorbs \(0.615 \mathrm{~kJ}\) of heat from its surroundings and has \(0.247 \mathrm{kcal}\) of work done on it?

Short answer

The change in internal energy is 1649.848 J.

Step-by-step solution

Convert Given Values to the Same Unit

First, convert all given values to the same unit. Here, we will convert everything to Joules. The system absorbs 0.615 kJ of heat: 0.615 kJ * 1000 J/kJ = 615 J. It has 0.247 kcal of work done on it: 0.247 kcal * 4184 J/kcal = 1034.848 J.

Apply the First Law of Thermodynamics

The first law of thermodynamics states that the change in internal energy \(\triangle U\) of a system is the sum of the heat \(q\) added to the system and the work \(w\) done on the system:ewline\(\triangle U = q + w\)

Substitute the Values and Compute

Substitute the values of \(q\) and \(w\) into the equation:ewline\triangle U = 615 \ J + 1034.848 \ JewlineSolving this gives:ewline\triangle U = 1649.848 \ J.

Key concepts

First Law of Thermodynamics

The first law of thermodynamics is a fundamental principle in physics. It can be described using the formula: \(\triangle U = q + w\). Here, \(\triangle U\) represents the change in internal energy of a system. The term \(q\) stands for the heat absorbed by the system, and \(w\) represents the work done on the system.
This law essentially states that energy cannot be created or destroyed. Instead, it can only be transferred or converted from one form to another. When a system absorbs heat (\(q > 0\)) or has work done on it (\(w > 0\)), its internal energy increases. Conversely, if a system releases heat or does work on its surroundings, its internal energy decreases.
Understanding this concept is crucial for solving problems related to energy changes within a physical system.

Unit Conversion to Joules

In physics, it is often necessary to convert different units of energy to a consistent unit like Joules (J) for straightforward calculations.
For instance, to convert kilojoules (kJ) to Joules, use the conversion factor: \(1 \text{kJ} = 1000 \text{J}\). Similarly, to convert kilocalories (kcal) to Joules, use: \(1 \text{kcal} = 4184 \text{J}\).
In our example, we converted 0.615 kJ to Joules: \(0.615 \text{kJ} \times 1000 \text{J/kJ} = 615 \text{J}\). We also converted 0.247 kcal to Joules: \(0.247 \text{kcal} \times 4184 \text{J/kcal} = 1034.848 \text{J}\).
Accurate unit conversion is essential for combining different energy quantities.

Heat Absorption

Heat absorption refers to the process where a system takes in thermal energy from its surroundings. This absorbed heat can change the internal energy of the system.
In the context of the first law of thermodynamics, absorbed heat (\(q\)) contributes positively to the change in internal energy. In our exercise, the system absorbs 0.615 kJ of heat. After conversion, this amount becomes 615 J.
Heat energy typically flows from a warmer object to a cooler one until thermal equilibrium is reached. The amount of heat absorbed or released can be calculated using various formulas depending on the scenario, such as \(q = m \times c \times \triangle T\), where \(m\) is mass, \(c\) is specific heat capacity, and \(\triangle T\) is the temperature change.

Work Done on System

Work done on a system implies energy transfer to the system through mechanical means, such as compression or expansion. In thermodynamics, positive work (\(w > 0\)) increases the system's internal energy.
During our exercise, 0.247 kcal of work is done on the system. Converting this to Joules, we get \(0.247 \text{kcal} \times 4184 \text{J/kcal} = 1034.848 \text{J}\).
Often, work done on a system is calculated through formulas like \(w = -P \times \triangle V\), where \(P\) is pressure and \(\triangle V\) is the change in volume. When work is done on the system (like compressing a gas), energy is added to the system, increasing its internal energy as per the first law of thermodynamics.