First Law of Thermodynamics for Closed Systems

Introduction: The basic equations for thermodynamics is used to solve for the majority of thermodynamic problems.

The Essentials

The first step for many thermodynamic problems is the First Law of Thermodynamics. This law can come in many forms for different types of systems, but all forms are based on the basic idea of energy conservation. Energy conservation states that the change in energy of a system is equal to the energy entering the system minus the energy leaving the system ( $\Delta E = E_{in}-E{out}$).

Closed systems (no mass transfer) are the easiest systems in which the First Law can be applied. With no mass transfer, the only forms of energy transfer are heat transfer and work. Thus, the First Law can be written as shown below. $$\Delta E = \Delta Q + \Delta W=Q_{in}-Q_{out}+W_{in}-W_{out}$$ This is written in extensive form, but can be written in intensive form as shown below. $$q_{in}-q_{out}+w_{in}-w_{out}=\Delta ke+\Delta pe +\Delta u$$ Commonly, it is assumed that for closed systems there is no change in kinetic or potential energy. Additionally, if the system is adiabatic or well insulated, there is no heat transfer and the first law simplifies to $w_{in}-w_{out}=u_2-u_1$. A metric commonly used in conjunction with the first law is efficiency. The efficiency ($\eta$) of a system is a way to measure the effectiveness of the system. Efficiency is therefore defined as the ratio between the desired output and the required input. $$\eta=\frac{\text{desired output}}{\text{required input}}$$

Example

The engine of a tractor is capable of producing 153 kJ of power and receives 250 J of heat from the hot summer sun. If the exhaust of the engine removes 31 kJ of heat from the engine, how much heat must be supplied to the engine to maintain a steady state system? If the gasoline has a heating value of 40 MJ/kg and the engine can burn it with 81 percent efficiency, how much gasoline is needed to maintain a steady state for one second ?

To start solving this problem, answer the first the question using the First Law. Write out the First Law completely, and then cross out any terms that are known to be equal to zero. For this problem, there is work supplied ($W_{in}$), heat removed ($Q_{out}$, and heat added from both the sun ($Q_{in,sun}$ and the gasoline ($Q_{in,gas}$. Additionally, a steady state means that the system will have no change in energy as time progresses. $$\bcancel{\Delta E} =Q_{in,sun}+Q_{in,gas}-Q_{out}+\bcancel{W_{in}}-W_{out}$$ After canceling our terms and solving for the heat added from fuel, we have the following equation. $$Q_{in,gas}=Q_{out}+W_{out}-Q_{in,sun}$$ Given values can be input. Note that for many thermodynamic problems, it is customary to use kJ instead of J to better reflect the magnitude of the energy transfers. $$Q_{in,gas}=31\text{ kJ}+153\text{ kJ}-0.25\text{ kJ}=183.75\text{ kJ}$$ Now that the heat addition from gasoline is found, the next step is to find the quantity of gasoline needed. The first step is to apply the efficiency equation to find the total heat available in the gasoline. For this instance, the desired output is the heat input to the engine and the required output is the total potential heat stored in the gasoline $$\eta=\frac{Q_{in}}{Q_{in,source}},\rightarrow Q_{in,source}=\frac{Q_{in}}{\eta}=\frac{183.75}{0.81}=226.852\text{ kJ}$$

Next, determine the mass of gasoline needed using the heating value. This can be done through the following relation. $$Q=HHV*m,\rightarrow m=\frac{Q_{in,source}}{HHV}=\frac{226.852}{40,000}=0.00567\text{ kg}$$

Practice

1: What is the change in internal energy of a container of water at rest that has 400 J of heat added through an electric heater and loses 271 J of heat to the environment? Assume the container has no change in velocity or height.

2: A system receives 31 kJ of heat, loses 12 kJ of heat to the environment and does some amount of work. The total internal energy of the system changes by 4 kJ. How much work is done by the system and what is the efficiency of the system in creating work?

Solutions:
1: 129 J

2: 15 kJ, 48.387%

More Resources

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