The first law of thermodynamics, also known as the law of energy conservation, plays a crucial role in understanding energy transfers in physical systems. It states that energy cannot be created or destroyed; it can only be transformed from one form to another. This principle is foundational in various scientific and engineering fields, especially in thermodynamics. In this article, we will explore the first law of thermodynamics through practical problems to enhance understanding and application of the concept. We will cover diverse types of problems, step-by-step solutions, and helpful tips for mastering this essential law.
Understanding the First Law of Thermodynamics
The first law of thermodynamics is mathematically represented as:
\[
\Delta U = Q - W
\]
Where:
- \(\Delta U\) = change in internal energy of the system
- \(Q\) = heat added to the system
- \(W\) = work done by the system on the surroundings
This equation signifies that the change in internal energy of a closed system is equal to the heat supplied to the system minus the work done by the system.
Types of Problems Involving the First Law
Problems involving the first law of thermodynamics can be categorized into several types:
1. Constant Volume Processes
2. Isothermal Processes
3. Adiabatic Processes
4. Cyclic Processes
5. Heat Engines and Refrigerators
Each type presents unique scenarios that require a specific approach to solving them.
1. Constant Volume Processes
In a constant volume process, no work is done since the volume does not change. The equation simplifies as follows:
\[
\Delta U = Q
\]
Example Problem 1:
A gas is heated at constant volume, absorbing 500 J of heat. What is the change in internal energy of the gas?
Solution:
Since no work is done, we have:
\[
\Delta U = Q = 500 \, \text{J}
\]
Thus, the change in internal energy is 500 J.
2. Isothermal Processes
In an isothermal process for an ideal gas, the temperature remains constant. The work done is equal to the heat absorbed:
\[
\Delta U = 0 \quad \Rightarrow \quad Q = W
\]
Example Problem 2:
An ideal gas expands isothermally at 300 K, doing 200 J of work. How much heat is absorbed by the gas?
Solution:
Since the process is isothermal:
\[
Q = W = 200 \, \text{J}
\]
Thus, the heat absorbed by the gas is 200 J.
3. Adiabatic Processes
In an adiabatic process, there is no heat exchange with the surroundings. Therefore:
\[
Q = 0 \quad \Rightarrow \quad \Delta U = -W
\]
Example Problem 3:
A piston compresses a gas adiabatically, doing 150 J of work on it. What is the change in internal energy of the gas?
Solution:
Since \(Q = 0\):
\[
\Delta U = -W = -150 \, \text{J}
\]
Thus, the change in internal energy is -150 J, indicating that the internal energy of the gas increases due to work done on it.
4. Cyclic Processes
In a cyclic process, the system returns to its initial state. Hence, the change in internal energy over one complete cycle is zero:
\[
\Delta U = 0 \quad \Rightarrow \quad Q = W
\]
Example Problem 4:
A heat engine operates in a cyclic process, absorbing 600 J of heat from a hot reservoir and performing 400 J of work. What is the heat rejected to the cold reservoir?
Solution:
From the first law:
\[
Q = W + Q_{out}
\]
Rearranging gives:
\[
Q_{out} = Q - W = 600 \, \text{J} - 400 \, \text{J} = 200 \, \text{J}
\]
Thus, the heat rejected to the cold reservoir is 200 J.
5. Heat Engines and Refrigerators
Heat engines convert heat into work, while refrigerators do the opposite. The efficiency of a heat engine is given by:
\[
\eta = \frac{W}{Q_{in}}
\]
Where \(W\) is the work done by the engine and \(Q_{in}\) is the heat absorbed.
Example Problem 5:
A heat engine absorbs 800 J of heat and performs 200 J of work. Calculate its efficiency.
Solution:
Using the efficiency formula:
\[
\eta = \frac{W}{Q_{in}} = \frac{200 \, \text{J}}{800 \, \text{J}} = 0.25
\]
Thus, the efficiency of the heat engine is 25%.
Solving Practice Problems Effectively
When tackling practice problems related to the first law of thermodynamics, consider the following steps:
1. Understand the Problem: Read the problem carefully to identify what is given and what needs to be found.
2. Identify the Process Type: Determine whether the process is isothermal, adiabatic, cyclic, or constant volume.
3. Apply the First Law: Use the appropriate form of the first law based on the identified process.
4. Perform Calculations: Carefully perform the necessary calculations, ensuring units are consistent.
5. Check Your Work: Review your calculations and results to confirm they make physical sense.
Conclusion
The first law of thermodynamics is a fundamental principle that governs energy interactions in various systems. Understanding and applying this law through practical problems is essential for students and professionals in fields such as physics, engineering, and chemistry. By practicing different types of problems, from constant volume processes to heat engines, one can develop a strong grasp of thermodynamic concepts. Emphasizing the problem-solving steps will further enhance your ability to tackle complex thermodynamic scenarios confidently. As you continue to practice, remember that mastery of the first law not only aids academic pursuits but also has practical applications in real-world energy systems.
Frequently Asked Questions
What is the first law of thermodynamics?
The first law of thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another. It is often expressed as ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.
How do you calculate the work done by a gas in an isothermal process?
In an isothermal process, the work done by a gas can be calculated using the formula W = nRT ln(Vf/Vi), where n is the number of moles, R is the ideal gas constant, T is the temperature in Kelvin, Vf is the final volume, and Vi is the initial volume.
What is the significance of the internal energy change in the first law of thermodynamics?
The internal energy change (ΔU) represents the total energy change within a system due to heat transfer and work done. It is crucial for understanding how energy flows in thermodynamic processes.
How can the first law of thermodynamics be applied to a closed system undergoing a phase change?
In a closed system undergoing a phase change, the first law states that the heat absorbed (Q) during the phase change is equal to the change in internal energy (ΔU), since no work is done (W = 0). This can be expressed as ΔU = Q.
What is a common practice problem involving the first law of thermodynamics?
A common practice problem is: 'A gas in a piston expands isothermally, absorbing 500 J of heat from the surroundings. If it does 300 J of work during this process, what is the change in internal energy?' The solution involves applying ΔU = Q - W, yielding ΔU = 500 J - 300 J = 200 J.
How does the first law of thermodynamics apply to adiabatic processes?
In an adiabatic process, there is no heat exchange with the surroundings (Q = 0). Therefore, the first law simplifies to ΔU = -W, meaning that the change in internal energy is equal to the negative of the work done by the system.
What is a practical example of the first law of thermodynamics in everyday life?
A practical example is a car engine, where fuel combustion provides energy (heat) that does work on the pistons. The energy input from the fuel results in a change in internal energy and performs mechanical work, demonstrating the first law.
Can the first law of thermodynamics predict the efficiency of a heat engine?
Yes, the first law of thermodynamics helps in calculating the efficiency of a heat engine by comparing the work output to the heat input. Efficiency (η) can be expressed as η = W/Qh, where Qh is the heat absorbed from the hot reservoir.
What role does the first law of thermodynamics play in refrigeration?
In refrigeration, the first law of thermodynamics explains how work is done on the refrigerant to transfer heat from a cold space to a hot space, using the formula Qc = W + Qh, which outlines the relationship between the heat removed, work input, and heat rejected.
