1. The problem statement, all variables and given/known data
Consider a constant volume elastic string. A change in internal energy of the string is given by dU = TdS + fdL.
The elastic string obeys the following two equations; $$\left(\frac{\partial f}{\partial L}\right)_L = \gamma_1 T\,\,\,\,\,\,\,\,\,\,\,\left(\frac{\partial f}{\partial T}\right)_L = \gamma_2L,$$ ##\gamma_1, \gamma_2## constants.
A) Why do we require ##\gamma_1 = \gamma_2##
B)Derive an expression for f(T,L). Explain whether this expression gives a complete thermodynamic description of the system.
2. Relevant equations
Total differential
3. The attempt at a solution
A) It is not exactly clear to me why the constants have to be the same. I was thinking that since L and T are independent variables that ##\gamma_1, \gamma_2## may be interpreted as separation constants, but I am unsure.
B)If ##df = \gamma_1 T dL + \gamma_2 L dT##, then I am tempted to write ##\Delta f = \gamma_2 T \Delta L + \gamma_2 L \Delta T##, but I don't think this is correct since L ( and T) are not state variables.
I think the resulting expression will give a complete thermodynamic description since we have expressed f as a function of two independent variables, and we have specified what is being held constant in each case.
Thanks.
Consider a constant volume elastic string. A change in internal energy of the string is given by dU = TdS + fdL.
The elastic string obeys the following two equations; $$\left(\frac{\partial f}{\partial L}\right)_L = \gamma_1 T\,\,\,\,\,\,\,\,\,\,\,\left(\frac{\partial f}{\partial T}\right)_L = \gamma_2L,$$ ##\gamma_1, \gamma_2## constants.
A) Why do we require ##\gamma_1 = \gamma_2##
B)Derive an expression for f(T,L). Explain whether this expression gives a complete thermodynamic description of the system.
2. Relevant equations
Total differential
3. The attempt at a solution
A) It is not exactly clear to me why the constants have to be the same. I was thinking that since L and T are independent variables that ##\gamma_1, \gamma_2## may be interpreted as separation constants, but I am unsure.
B)If ##df = \gamma_1 T dL + \gamma_2 L dT##, then I am tempted to write ##\Delta f = \gamma_2 T \Delta L + \gamma_2 L \Delta T##, but I don't think this is correct since L ( and T) are not state variables.
I think the resulting expression will give a complete thermodynamic description since we have expressed f as a function of two independent variables, and we have specified what is being held constant in each case.
Thanks.
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