# Is the difference of two state functions a state function?

Tags:
1. Apr 6, 2015

### tomkoolen

Hello everybody,

For my thermodynamics test I have to tell whether or not a quantity is a state function, which is obviously not all too difficult when regarding entropy, enthalpy etc. on their own. However there are a lot of questions where it is about "H-S" or "G-H". Are these not always state functions as well?

2. Apr 6, 2015

### MarcusAgrippa

Clue: what is the definition of a state function? And what do you get when you form H-S, etc?

3. Apr 6, 2015

### tomkoolen

I understand that a state function defines the state of a system regardless of the path by which the state was obtained. Both H and S are state functions but I don't get the meaning of the difference function, on intuition I would say that it is a state function as well. Can anybody verify?

4. Apr 7, 2015

### MarcusAgrippa

That is not the answer. A state function does not define the state of a system. You need to define quantities properly.

The state of a system is defined by the values of the largest number of independent state variables for the system. You are probably at the moment only considering the simplest type of thermodynamic system, called a simple hydrostatic system. The equilibrium state of such a system is fully defined by specifying two of the three variables P, V and T. The variables P, V and T are not independent variables. they are related to each other by an equation, called the equation of state for the system. Three variable related by one equation means that only two can be assigned values independently. The value of the third is determined from these two values by the equation of state. This is expressed by saying that two of the variables are independent, and the third is dependent. So the state of a system is defined by specifying the values of T and V, or of T and P, or of P and V.

A state function is a quantity that is a function of the variables that define the state of a system. Suppose we choose to define the equilibrium state of the system by the values of T and V. Then a state function is any function F of T and V, thus any F = F(T,V) is a state function.

Now, H is a state function, therefore H = H(T,V). Also S is a state function, therefore S = S(T,V). Subtracting S from H gives H(T,V) - S(T,V) = K(T,V) which is also a function of the state variables T and V. So H - S is therefore also a state variable.

Note how this was PROVED. Begin from the definition of the thing in question. Then use the definition of the other quantity to show that it fits the definition of that thing. You cannot prove anything without using definitions in conjunction with known proven results. So make sure that you memorise all definitions. You will not be able to do anything without them. Definitions are key information. Memorise all key information.

Remember that physics and mathematics are precise subjects. You need to develop precision in your thinking. Vague, wooly, touchy-feely "understanding" is not understanding at all.

You may be wondering if everything in TD is a state variable. The heat intake into a system is not a state function. Why? Because it depends on the process (path!) followed in going from the initial to the final equilibrium state. Similarly, the work done by the system is not a state variable.

Last edited: Apr 7, 2015
5. Apr 7, 2015

### Staff: Mentor

I'm not so sure I would consider H-S a state function. H and S don't even have the same units, so what would be the units of this strange new function? And what could something like H-S ever be used for? What do you guys think?

Chet

6. Apr 7, 2015

### MarcusAgrippa

Tomkoolen: please ignore #4 above and go directly to #10.

In #4, I carelessly assumed that H and S were used above to represent general functions of state variables - implicitly, of the same units, rather than entropy and enthalpy. So I have rewritten the answer to correct for this. The only point that I was trying to make in #4 was that a sum or difference of state functions (that are measured in the same units) is another state function.

Last edited: Apr 7, 2015
7. Apr 7, 2015

### Staff: Mentor

So you are saying that you were not necessarily interpreting H and S as enthalpy and entropy?

Chet

8. Apr 7, 2015

### MarcusAgrippa

Yes. Which admittedly is confusing in the context. So I shall edit the above comment to reflect this, if we are agreed on the rest of it.

9. Apr 7, 2015

### Staff: Mentor

Sounds good.

Chet

10. Apr 7, 2015

### MarcusAgrippa

THE FOLLOWING REPLACES #4, which I am no longer able to edit:

A state function does not define the state of a system. You need to define quantities properly.

The state of a system is defined by the values of the largest number of independent state variables for the system. You are probably at the moment only considering the simplest type of thermodynamic system, called a simple hydrostatic system. The equilibrium state of such a system is fully defined by specifying two of the three variables P, V and T. The variables P, V and T are not independent variables. they are related to each other by an equation, called the equation of state for the system. Three variable related by one equation means that only two can be assigned values independently. The value of the third is determined from these two values by the equation of state. This is expressed by saying that two of the variables are independent, and the third is dependent. So the state of a system is defined by specifying the values of T and V, or of T and P, or of P and V.

A state function f is a quantity that is a function of the variables that define the state of a system. Suppose we choose to define the equilibrium state of the system by the values of T and V. Then a state function f is any function of T and V. Thus any function f = f(T,V) is a state function.

In mathematics, the values of all variables are dimensionless. Their values are pure numbers. We have an added complication in physics: the variables usually are not dimensionless. For example T is measured in Kelvin, and V is measured in cubic metres. So all expressions that we form must be dimensionally consistent as well as mathematically correct.

If f and g are state variables measured in the same units, it makes sense to add or subtract f and g. Their sum is then f(T,V) + g(T,V). This sum is also a function h(T,V) of the state variables T and V, and so is a state function. If their dimensions are different, however, it makes no sense to add or subtract them. However, you could multiply or divide them to get a new state variable with dimensions different from both f and g.

Enthalpy is a state function. Therefore H = H(T,V). Also entropy is a state function. Therefore S = S(T,V). However, they are measured in different units: S is measured in Joules per Kelvin, while H is measured in Joules. So even though they are each state functions, adding them is not dimensionally consistent. It makes no sense to add or subtract them. This means that you cannot form a new state variable by adding them or subtracting them. You could however form a new state variable by multiplying or dividing them.

Since S is measured in Joules per Kelvin, and T is measured in Kelvin, you could form the product TS. The product TS is then a function of T and V and so is a state variable. It is measured in Joules. H is also measured in Joules, so you could form a new state variable by taking TS + H or TS - H.

Note how the above was PROVED. Begin from the definition of the thing in question. Then use the definition of the other quantity to show that it fits the definition of that thing. You cannot prove anything without using definitions in conjunction with known proven results. So make sure that you memorise all definitions. You will not be able to do anything without them. Definitions are key information. Memorise all key information. When doing physics, you have the additional constraint that every expression or equation that you write down must be dimensionally correct. So always also check all the dimensions.

Remember that physics and mathematics are precise subjects. You need to develop precision in your thinking. Vague, wooly, touchy-feely "understanding" is not understanding at all.

You may be wondering if everything in thermodynamics is a state variable. The heat intake into a system is not a state function. Why? Because it depends on the process (path!) followed in going from the initial to the final equilibrium state. Similarly, the work done by the system is not a state function.

Last edited: Apr 7, 2015
11. Apr 7, 2015

### MarcusAgrippa

Chet, won't you check what I have written to make sure that it is now saying what it ought?

12. Apr 7, 2015

### Staff: Mentor

Yes. Looks good.

Chet