How to solve constants out of the internal energy equation?Isothermal vs. adiabatic compression of gas in terms of required energyCorrelation between the virial coefficients and a & b in the corresponding Van Der Waals equation of stateHow to derive the pressure dependency for the Gibbs free energy?How to find and use the Clausius-Clapeyron equationWhy is Gibbs free energy more useful than internal energy?Average or individual molar heat capacity?How does one solve the following differential equation (mass balance equation)Calculations on an irreversible adiabatic expansionWhy does more heat transfer take place in a reversible process than in a irreversible process?Deriving heat capacity in terms of internal energy U and natural variables S & V
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How to solve constants out of the internal energy equation?
Isothermal vs. adiabatic compression of gas in terms of required energyCorrelation between the virial coefficients and a & b in the corresponding Van Der Waals equation of stateHow to derive the pressure dependency for the Gibbs free energy?How to find and use the Clausius-Clapeyron equationWhy is Gibbs free energy more useful than internal energy?Average or individual molar heat capacity?How does one solve the following differential equation (mass balance equation)Calculations on an irreversible adiabatic expansionWhy does more heat transfer take place in a reversible process than in a irreversible process?Deriving heat capacity in terms of internal energy U and natural variables S & V
$begingroup$
Imagine we deal with a new kind of matter, whose state is described by:
$$PV = AT^3$$
Its internal energy is given by:
$$U = BT^n lnleft(fracVV_0right) + f(T)$$
Where $A, B$ and $V_0$ is a constant and $f(T)$ is a polynomial function.
Find B and n.
This is what I know:
The given expressions remind me of adiabatic compression/expansion. If we assume quasistatic adiabatic compression/expansion we know that heat won't get out/in the system.
$$Delta U = -W$$
And work is:
$$W = -PDelta V$$
Some thoughts on how to solve the problem
We notice here that we are dealing with a non-ideal gas. Assuming that the above equations are correct and using first thermodynamics law one gets:
$$mathrmdU = [nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT$$
$$[nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT = fracAT^3VmathrmdV$$
I do not see a solution to this differential equation.
There has to be a easier way to get both $B$ and $n$ but how?
Thanks.
thermodynamics equation-of-state
edited 1 hour ago
Gaurang Tandon
5,46262864
asked 1 hour ago
JD_PMJD_PM
1726
$endgroup$
add a comment |
$begingroup$
Imagine we deal with a new kind of matter, whose state is described by:
$$PV = AT^3$$
Its internal energy is given by:
$$U = BT^n lnleft(fracVV_0right) + f(T)$$
Where $A, B$ and $V_0$ is a constant and $f(T)$ is a polynomial function.
Find B and n.
This is what I know:
The given expressions remind me of adiabatic compression/expansion. If we assume quasistatic adiabatic compression/expansion we know that heat won't get out/in the system.
$$Delta U = -W$$
And work is:
$$W = -PDelta V$$
Some thoughts on how to solve the problem
We notice here that we are dealing with a non-ideal gas. Assuming that the above equations are correct and using first thermodynamics law one gets:
$$mathrmdU = [nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT$$
$$[nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT = fracAT^3VmathrmdV$$
I do not see a solution to this differential equation.
There has to be a easier way to get both $B$ and $n$ but how?
Thanks.
thermodynamics equation-of-state
edited 1 hour ago
Gaurang Tandon
5,46262864
asked 1 hour ago
JD_PMJD_PM
1726
$endgroup$
1
$begingroup$
Your differential equation doesn't make sense to me: you have infinitesimals ($mathrm dU$ and $mathrm df$) and finite quantities ($nBT^n-1ln(V/V_0)$) being added together. I gather you were trying to differentiate by $T$ throughout?
$endgroup$
– orthocresol♦
1 hour ago
$begingroup$
@orthocresol that is a typo let me fix it.
$endgroup$
– JD_PM
1 hour ago
$begingroup$
Yes I differentiated $U$ with respect to $T$. The idea is to set up a differential equation that relates the change in temperature and volume during the compression/expansion process. I assumed it will be adiabatic (based on the given equation: $PV = AT^3$)
$endgroup$
– JD_PM
1 hour ago
$begingroup$
By using the equation $ big( fracpartial Upartial Vbig)_T = Tbig(fracpartial Ppartial Tbig)_V - P $, you will get $B$ as $2A$ and $n=3$.
$endgroup$
– Soumik Das
49 mins ago
add a comment |
$begingroup$
Imagine we deal with a new kind of matter, whose state is described by:
$$PV = AT^3$$
Its internal energy is given by:
$$U = BT^n lnleft(fracVV_0right) + f(T)$$
Where $A, B$ and $V_0$ is a constant and $f(T)$ is a polynomial function.
Find B and n.
This is what I know:
The given expressions remind me of adiabatic compression/expansion. If we assume quasistatic adiabatic compression/expansion we know that heat won't get out/in the system.
$$Delta U = -W$$
And work is:
$$W = -PDelta V$$
Some thoughts on how to solve the problem
We notice here that we are dealing with a non-ideal gas. Assuming that the above equations are correct and using first thermodynamics law one gets:
$$mathrmdU = [nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT$$
$$[nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT = fracAT^3VmathrmdV$$
I do not see a solution to this differential equation.
There has to be a easier way to get both $B$ and $n$ but how?
Thanks.
thermodynamics equation-of-state
edited 1 hour ago
Gaurang Tandon
5,46262864
asked 1 hour ago
JD_PMJD_PM
1726
$endgroup$
Imagine we deal with a new kind of matter, whose state is described by:
$$PV = AT^3$$
Its internal energy is given by:
$$U = BT^n lnleft(fracVV_0right) + f(T)$$
Where $A, B$ and $V_0$ is a constant and $f(T)$ is a polynomial function.
Find B and n.
This is what I know:
The given expressions remind me of adiabatic compression/expansion. If we assume quasistatic adiabatic compression/expansion we know that heat won't get out/in the system.
$$Delta U = -W$$
And work is:
$$W = -PDelta V$$
Some thoughts on how to solve the problem
We notice here that we are dealing with a non-ideal gas. Assuming that the above equations are correct and using first thermodynamics law one gets:
$$mathrmdU = [nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT$$
$$[nBT^n-1 lnleft(fracVV_0right) + f'(T)]mathrmdT = fracAT^3VmathrmdV$$
I do not see a solution to this differential equation.
There has to be a easier way to get both $B$ and $n$ but how?
Thanks.
thermodynamics equation-of-state
thermodynamics equation-of-state
edited 1 hour ago
Gaurang Tandon
5,46262864
asked 1 hour ago
JD_PMJD_PM
1726
edited 1 hour ago
Gaurang Tandon
5,46262864
asked 1 hour ago
JD_PMJD_PM
1726
edited 1 hour ago
Gaurang Tandon
5,46262864
edited 1 hour ago
Gaurang Tandon
5,46262864
edited 1 hour ago
Gaurang Tandon
5,46262864
5,46262864
asked 1 hour ago
JD_PMJD_PM
1726
asked 1 hour ago
JD_PMJD_PM
1726
asked 1 hour ago
JD_PMJD_PM
1726
1726
1
$begingroup$
Your differential equation doesn't make sense to me: you have infinitesimals ($mathrm dU$ and $mathrm df$) and finite quantities ($nBT^n-1ln(V/V_0)$) being added together. I gather you were trying to differentiate by $T$ throughout?
$endgroup$
– orthocresol♦
1 hour ago
$begingroup$
@orthocresol that is a typo let me fix it.
$endgroup$
– JD_PM
1 hour ago
$begingroup$
Yes I differentiated $U$ with respect to $T$. The idea is to set up a differential equation that relates the change in temperature and volume during the compression/expansion process. I assumed it will be adiabatic (based on the given equation: $PV = AT^3$)
$endgroup$
– JD_PM
1 hour ago
$begingroup$
By using the equation $ big( fracpartial Upartial Vbig)_T = Tbig(fracpartial Ppartial Tbig)_V - P $, you will get $B$ as $2A$ and $n=3$.
$endgroup$
– Soumik Das
49 mins ago
add a comment |
1
$begingroup$
Your differential equation doesn't make sense to me: you have infinitesimals ($mathrm dU$ and $mathrm df$) and finite quantities ($nBT^n-1ln(V/V_0)$) being added together. I gather you were trying to differentiate by $T$ throughout?
$endgroup$
– orthocresol♦
1 hour ago
$begingroup$
@orthocresol that is a typo let me fix it.
$endgroup$
– JD_PM
1 hour ago
$begingroup$
Yes I differentiated $U$ with respect to $T$. The idea is to set up a differential equation that relates the change in temperature and volume during the compression/expansion process. I assumed it will be adiabatic (based on the given equation: $PV = AT^3$)
$endgroup$
– JD_PM
1 hour ago
$begingroup$
By using the equation $ big( fracpartial Upartial Vbig)_T = Tbig(fracpartial Ppartial Tbig)_V - P $, you will get $B$ as $2A$ and $n=3$.
$endgroup$
– Soumik Das
49 mins ago
1
1
$begingroup$
Your differential equation doesn't make sense to me: you have infinitesimals ($mathrm dU$ and $mathrm df$) and finite quantities ($nBT^n-1ln(V/V_0)$) being added together. I gather you were trying to differentiate by $T$ throughout?
$endgroup$
– orthocresol♦
1 hour ago
$begingroup$
Your differential equation doesn't make sense to me: you have infinitesimals ($mathrm dU$ and $mathrm df$) and finite quantities ($nBT^n-1ln(V/V_0)$) being added together. I gather you were trying to differentiate by $T$ throughout?
$endgroup$
– orthocresol♦
1 hour ago
$begingroup$
@orthocresol that is a typo let me fix it.
$endgroup$
– JD_PM
1 hour ago
$begingroup$
@orthocresol that is a typo let me fix it.
$endgroup$
– JD_PM
1 hour ago
$begingroup$
Yes I differentiated $U$ with respect to $T$. The idea is to set up a differential equation that relates the change in temperature and volume during the compression/expansion process. I assumed it will be adiabatic (based on the given equation: $PV = AT^3$)
$endgroup$
– JD_PM
1 hour ago
$begingroup$
Yes I differentiated $U$ with respect to $T$. The idea is to set up a differential equation that relates the change in temperature and volume during the compression/expansion process. I assumed it will be adiabatic (based on the given equation: $PV = AT^3$)
$endgroup$
– JD_PM
1 hour ago
$begingroup$
By using the equation $ big( fracpartial Upartial Vbig)_T = Tbig(fracpartial Ppartial Tbig)_V - P $, you will get $B$ as $2A$ and $n=3$.
$endgroup$
– Soumik Das
49 mins ago
$begingroup$
By using the equation $ big( fracpartial Upartial Vbig)_T = Tbig(fracpartial Ppartial Tbig)_V - P $, you will get $B$ as $2A$ and $n=3$.
$endgroup$
– Soumik Das
49 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You need to use the equation $$left(fracpartial Upartial Vright)_T=-left[P-Tleft(fracpartial Ppartial Tright)_Vright]$$
answered 52 mins ago
Chet MillerChet Miller
6,7561713
$endgroup$
add a comment |
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$begingroup$
You need to use the equation $$left(fracpartial Upartial Vright)_T=-left[P-Tleft(fracpartial Ppartial Tright)_Vright]$$
answered 52 mins ago
Chet MillerChet Miller
6,7561713
$endgroup$
add a comment |
$begingroup$
You need to use the equation $$left(fracpartial Upartial Vright)_T=-left[P-Tleft(fracpartial Ppartial Tright)_Vright]$$
answered 52 mins ago
Chet MillerChet Miller
6,7561713
$endgroup$
add a comment |
$begingroup$
You need to use the equation $$left(fracpartial Upartial Vright)_T=-left[P-Tleft(fracpartial Ppartial Tright)_Vright]$$
answered 52 mins ago
Chet MillerChet Miller
6,7561713
$endgroup$
You need to use the equation $$left(fracpartial Upartial Vright)_T=-left[P-Tleft(fracpartial Ppartial Tright)_Vright]$$
answered 52 mins ago
Chet MillerChet Miller
6,7561713
answered 52 mins ago
Chet MillerChet Miller
6,7561713
answered 52 mins ago
Chet MillerChet Miller
6,7561713
answered 52 mins ago
Chet MillerChet Miller
6,7561713
6,7561713
add a comment |
add a comment |
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$begingroup$
Your differential equation doesn't make sense to me: you have infinitesimals ($mathrm dU$ and $mathrm df$) and finite quantities ($nBT^n-1ln(V/V_0)$) being added together. I gather you were trying to differentiate by $T$ throughout?
$endgroup$
– orthocresol♦
1 hour ago
$begingroup$
@orthocresol that is a typo let me fix it.
$endgroup$
– JD_PM
1 hour ago
$begingroup$
Yes I differentiated $U$ with respect to $T$. The idea is to set up a differential equation that relates the change in temperature and volume during the compression/expansion process. I assumed it will be adiabatic (based on the given equation: $PV = AT^3$)
$endgroup$
– JD_PM
1 hour ago
$begingroup$
By using the equation $ big( fracpartial Upartial Vbig)_T = Tbig(fracpartial Ppartial Tbig)_V - P $, you will get $B$ as $2A$ and $n=3$.
$endgroup$
– Soumik Das
49 mins ago