# Proving Specific Enthelpy = Specific Internal Energy

• v_pino
In summary, the conversation discusses the relationship between specific enthalpy and specific internal energy for an ideal gas at constant pressure. The individual is seeking proof for this relationship and provides equations to support their understanding. They also ask for clarification on how to obtain specific internal energy and how to proceed once it is obtained.
v_pino
I read that for an ideal gas, the specific enthalpy (h) at constant pressure is equal to the specific internal energy of the gas. How do I proof that?

I know that:

1.) specific enthalpy = specific internal energy + pressure x specific volume

2.) Internal energy at constant pressure = mass of gas x capacity x change in tempt. + (pressure x volume of gas)

3.) pressure x specific volume = gas constant x temperature

How do I obtain specific internal energy? Do I simply divide both sides by (m)?

How do I proceed on after I've obtained specific internal energy?

v_pino said:
I read that for an ideal gas, the specific enthalpy (h) at constant pressure is equal to the specific internal energy of the gas.

Where did you read this? It doesn't sound right, as written.

I understand your curiosity in proving the relationship between specific enthalpy and specific internal energy for an ideal gas. To prove this relationship, we can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. For an ideal gas, the change in internal energy can be expressed as:

ΔU = mCΔT - PΔV

Where m is the mass of the gas, C is the specific heat capacity, ΔT is the change in temperature, P is the pressure, and ΔV is the change in volume.

Now, let's consider a constant pressure process, where the pressure remains constant throughout the process. In this case, the change in internal energy can be simplified to:

ΔU = mCΔT - PΔV = mCΔT - PΔV = mCΔT - PΔV = mCΔT - PΔV

Since the pressure remains constant, we can also express the change in enthalpy as:

ΔH = mCΔT + PΔV

Comparing the two equations, we can see that the change in enthalpy is equal to the change in internal energy plus the work done by the system, which is equal to PΔV. This is consistent with the definition of enthalpy as the sum of internal energy and the product of pressure and volume.

Therefore, at constant pressure, the specific enthalpy and specific internal energy can be expressed as:

h = u + Pv

Where h is the specific enthalpy, u is the specific internal energy, P is the pressure, and v is the specific volume.

To obtain the specific internal energy, you can simply divide both sides by the mass of the gas (m):

u = (h - Pv)/m

This equation shows that the specific internal energy is equal to the specific enthalpy minus the product of pressure and specific volume, all divided by the mass of the gas.

In summary, the relationship between specific enthalpy and specific internal energy for an ideal gas at constant pressure can be proven using the first law of thermodynamics and the definition of enthalpy. I hope this explanation helps in your understanding.

## What is specific enthalpy and specific internal energy?

Specific enthalpy and specific internal energy are thermodynamic properties that describe the energy content of a substance. Specific enthalpy is the total energy of a substance, including both its internal energy and the energy required to overcome its surrounding pressure, while specific internal energy is the energy contained within the substance's molecules.

## How do you prove that specific enthalpy equals specific internal energy?

The proof for this equation is based on the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. By considering a substance at constant pressure and using the definition of enthalpy, we can show that the change in specific enthalpy is equal to the change in specific internal energy plus the product of pressure and volume.

## Why is it important to prove this relationship between specific enthalpy and specific internal energy?

This relationship is important because it allows us to simplify thermodynamic calculations and make predictions about the behavior of substances. It also helps us understand the energy transfer within a system and how changes in pressure and volume can affect the enthalpy and internal energy of a substance.

## Can this relationship be applied to all substances?

Yes, this relationship holds true for all substances, regardless of their chemical composition or physical state. However, it is important to note that the specific enthalpy and specific internal energy values may vary depending on the substance and its properties.

## Are there any limitations to this relationship?

While this relationship is generally applicable, there are some cases where it may not hold true. For example, it may not be accurate for substances undergoing phase changes, or for substances experiencing extreme temperatures or pressures. In these cases, more complex thermodynamic equations may need to be used to accurately describe the behavior of the substance.

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