When to use C(subv) and C(subp) for Q heat equation

In summary: The summary of this conversation is that the internal energy of the air inside a basketball changes when the ball is compressed to 80.5% of its original volume. This change can be calculated using the equation ##\Delta(U)=nC_v\Delta(T)##, where ##C_v## is the specific heat at constant volume. The use of ##C_v## is appropriate for this adiabatic process, even though the volume of the gas changes. The heat transfer, ##Q##, in this process is equal to zero.
  • #1
nso09
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Homework Statement


A player bounces a basketball on the floor, compressing it to 80.5% of its original volume. The air (assume it is essentially N2 gas) inside the ball is originally at a temperature of 20.5°C and a pressure of 1.80 atm. The ball's diameter is 23.9 cm.
By how much does the internal energy of the air change between the ball's original state and its maximum compression?
Given: initial and final volume, initial temperature 20.5 degrees celsius, initial pressure 1.80 atm, diameter

Homework Equations


##Q=nC_v\Delta(T)##
##C_p=C_v+R##

The Attempt at a Solution


Q=0 since this seems to be an adiabatic process. Therefore ##W=-\Delta(U)##
So ##\Delta(U)=nC_v\Delta(T)##

My question is, why can't we use ##C_p## since the pressure is constant? I am confused on the cases where you have to use ##C_p## or ##C_v##. Is this just because it is adiabatic? The volume seems to change though since they tell us that the ball compresses to 80.5% of its original volume. So why ##C_v?##
 
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  • #2
##U## is a state variable. So the change, ##\Delta U##, when going from some initial state to some final state is independent of the type of process connecting the two states. For an ideal gas ##U = nC_V T##, which expresses that ##U## is proportional to ##T##. The proportionality constant happens to be ##nC_V##. But the ##C_V## here is just a number and it does not imply that you must have a constant volume process in order to use ##\Delta U = nC_V\Delta T##. This equation is valid for all processes for any ideal gas.

However, the heat ##Q## transferred in going from the initial state to the final state does depend on the particular process. ##Q = nC_V \Delta T## is used for a constant volume process, while ##Q = nC_P \Delta T## is for a constant pressure process.
 
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  • #3
nso09 said:

Homework Statement


A player bounces a basketball on the floor, compressing it to 80.5% of its original volume. The air (assume it is essentially N2 gas) inside the ball is originally at a temperature of 20.5°C and a pressure of 1.80 atm. The ball's diameter is 23.9 cm.
By how much does the internal energy of the air change between the ball's original state and its maximum compression?
Given: initial and final volume, initial temperature 20.5 degrees celsius, initial pressure 1.80 atm, diameter

Homework Equations


##Q=nC_v\Delta(T)##
##C_p=C_v+R##

The Attempt at a Solution


Q=0 since this seems to be an adiabatic process. Therefore ##W=-\Delta(U)##
So ##\Delta(U)=nC_v\Delta(T)##

My question is, why can't we use ##C_p## since the pressure is constant? I am confused on the cases where you have to use ##C_p## or ##C_v##. Is this just because it is adiabatic? The volume seems to change though since they tell us that the ball compresses to 80.5% of its original volume. So why ##C_v?##
See the following: http://physics.stackexchange.com/qu...delta-e-int-nc-v-delta-t-for-an/295200#295200
See also: https://www.physicsforums.com/threads/conditions-for-writing-q-cdt.882430/#post-5546416
 
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FAQ: When to use C(subv) and C(subp) for Q heat equation

What is C(subv) and C(subp) in the context of the Q heat equation?

C(subv) and C(subp) refer to the volumetric and surface specific heat capacities, respectively, in the Q heat equation. These values represent the amount of heat required to raise the temperature of a substance by a certain amount per unit volume or surface area.

When should I use C(subv) instead of C(subp) in the Q heat equation?

C(subv) should be used when calculating heat transfer in a volumetric system, such as a solid or liquid. C(subp) is more appropriate for calculating heat transfer at a surface, such as in a gas or thin film.

Can I use either C(subv) or C(subp) interchangeably in the Q heat equation?

No, C(subv) and C(subp) represent different properties of a substance and should not be used interchangeably. Using the wrong value can result in inaccurate calculations.

How do I determine the values for C(subv) and C(subp) for a specific material?

The values for C(subv) and C(subp) can be found in reference tables or can be calculated using the material's specific heat capacity and density. These values may also vary depending on temperature and other factors, so it is important to use the most accurate values available.

Are there any other factors that may affect the use of C(subv) and C(subp) in the Q heat equation?

Yes, factors such as thermal conductivity, heat transfer coefficient, and boundary conditions may also impact the use of C(subv) and C(subp) in the Q heat equation. It is important to consider all relevant variables when choosing which value to use.

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