Ideal Gases: dU=dW+dQ and dU=3/2RdT

[georg gill]

(I):

dU=dW+dQ

also (II):

$$dU=\frac{3}{2}RdT$$

if you compress a gas dW in dU is positive from pV=nRT lesser volume could either mean more pressure or more T. If dV gives dp only then dT=0 how can then dU for dW in (I) be equal dU in (II)?

 

[MrSid]

George; don't confuse your general equation of state (I) with the solved equation of state with respect to the setting of a constant T, p, or V or adiabatic condition (II).

What is implied is the question of what term in the general equation goes to zero when the solved equation is presented for a process; then using the conditions that create that solution, allows one to use the ideal gas law to show relationships of the variables that are not being held constant.

Do you see where this is going now?

 

[georg gill]

George; don't confuse your general equation of state (I) with the solved equation of state with respect to the setting of a constant T, p, or V or adiabatic condition (II).

What is implied is the question of what term in the general equation goes to zero when the solved equation is presented for a process; then using the conditions that create that solution, allows one to use the ideal gas law to show relationships of the variables that are not being held constant.

Do you see where this is going now?

I am a bit confused unfortunately. Why is (II) adiabatic?

thank you

 

[MrSid]

sorry George- not to imply that (II) is adiabatic;

but just to set the stage that one of those four conditions as constant when the equation of state is solved- In fact you will find in the wiki article Table_of_thermodynamic_equations
about halfway down an Equation Table for ideal gas and for the thermodynamic equation that matches (II) the variable that is kept constant is volume (isochoric).

 

[LudusRex]

In (I), the equation dU=dW+dQ represents the change in internal energy (dU) of an ideal gas when work (dW) and heat (dQ) are applied to it. This equation takes into account both the change in volume (V) and the change in temperature (T) of the gas.

In (II), the equation dU=\frac{3}{2}RdT represents the change in internal energy (dU) of an ideal gas solely due to a change in temperature (dT). This equation is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat (dQ) added to the system minus the work (dW) done by the system.

It is important to note that in both equations, the term dU represents the total change in internal energy, which includes both the change in temperature and the work done on the gas. Therefore, even if dT=0, there can still be a change in internal energy if work is done on the gas, as shown in (I).

In summary, both equations (I) and (II) are valid and represent different aspects of the change in internal energy of an ideal gas. The first equation takes into account both work and heat, while the second equation focuses solely on the change in temperature.