Show that Tv^(R/c_v) = constant

Thread starter romanski007
Start date May 20, 2020
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In summary, the equation Tv^(R/c_v) = constant, also known as the adiabatic equation, is used to describe the relationship between temperature, volume, and specific heat in a closed system. It is derived from the first law of thermodynamics and the constant in the equation represents the adiabatic index, which is unique for each type of gas and can be used to analyze the behavior of gases under different conditions. While it can be applied to all types of gases, it is most useful for adiabatic processes and has practical applications in various fields such as engine design, weather analysis, and astrophysics.

May 20, 2020

romanski007

Homework Statement: The equation of state of a certain gas (P+b)v=RT and its specific internal energy u is given by u=aT+bv+u_0 where a ,b, u_0 and R are constants.
a) Find c_v (DONE)
b) Show that c_p - c_v =R for this gas (DONE)
c) Using the equation in (b) show that Tv^(R/c_v) = constant

Relevant Equations: (P+b)v=RT
u=aT+bv+u_0

The solutions start from the fact that c_v= (dT/dV)_s = -[(du/dv)_T + P], however I cannot reason where did that come from. Any help will be appreciated.

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May 20, 2020

Chestermiller

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$$du=Tds-Pdv=\frac{\partial u}{\partial T}dT+\frac{\partial u}{\partial v}dv=C_vdT+\frac{\partial u}{\partial v}dv$$so
$$Tds-Pdv=C_vdT+\frac{\partial u}{\partial v}dv$$

1. What does the equation Tv^(R/c_v) = constant represent?

The equation represents the relationship between temperature (T), volume (V), specific gas constant (R), and specific heat capacity at constant volume (c_v). It states that when the product of temperature and v^(R/c_v) is held constant, the value of Tv^(R/c_v) will remain constant.

2. How is this equation derived?

This equation is derived from the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant, and T is temperature. By rearranging the equation to isolate T and substituting c_v = R/(n-1), we can arrive at the equation Tv^(R/c_v) = constant.

3. What is the significance of this equation in thermodynamics?

This equation is significant in thermodynamics as it relates the fundamental properties of temperature, volume, and specific heat capacity in a closed system. It allows us to understand how changes in these properties affect each other, and how they can be controlled to maintain a constant value.

4. Can this equation be applied to all types of gases?

Yes, this equation can be applied to all types of ideal gases, as long as the conditions of constant volume and constant number of moles are met. However, it may not accurately represent the behavior of real gases, which may deviate from ideal gas behavior under certain conditions.

5. How is this equation used in practical applications?

This equation is commonly used in engineering and thermodynamics to calculate and predict the behavior of gases in various systems. It can also be used to determine the specific heat capacity of a gas at constant volume, which is an important parameter in many industrial processes.