Calculate these 5 temperatures along this Thermodynamic cycle

In summary, the conversation discusses the relationship between temperatures T1, T2, T3, and T4 in a system with ideal gas and an inverse proportionality between pressure and volume. Using this information and equations, it is possible to calculate the temperature of T2 as 2T1, but the temperatures of A and C cannot be determined without more information. It is suggested that A and C are on the same isotherm and therefore have the same temperature.
  • #1
Seeit
5
0
Homework Statement
Calculate the temperatures at places 2, 4, A, B and C if you know:
It's an ideal diatomic gas
T3 = 4T1
T2 = Tb = T4
The axis connecting 1, B and 3 crosses zero.
Relevant Equations
pV = nRT
Laws of thermodynamics
Screenshot_20230325_165532_WPS Office.jpg

I only know T3 = 4•T1
I was able to calculate the T2 = Tb = T4
I built four equations:
T2 = p2V1 / nR
T4 = p1V2 / nR
p1/T1 = p2/T2
V1/T2 = V2/4T1

I put them together and got T2 = 2T1

I can't figure out the temperatures of A and C. I tend to think Ta could equal Tc (then I would be able to calculate it), but I am not sure.
 
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  • #2
Helo @Seeit ,
:welcome: ##\qquad## !​
Are you sure you have rendered the complete problem statement ? I would expect some more information, like: ideal gas, isothermal (or adiabatic), ...

I also miss an equation of state in your relevant equations (e.g. ##\ pV = NRT##).

Seeit said:
I only know ##T_3 = 4T_1##
How ? Or was that a given ? (In that case it is part of the problem statement)
Same for ##T_2 = T_b = T_4## ?

What about the scale and the axes of the diagram ?

##\ ##
 
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  • #3
Welcome, @Seeit !

As post #1 has been edited to answer @BvU questions, I suggest considering two things:

-The inverse proportionality between p and V.
-The similarity between polygons 1CBA and 1234 due to the axis connecting 1, B and 3, which makes their corresponding sides proportional.

direct-and-inverse-proportion-1629696427.png
 
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  • #4
Lnewqban said:
Welcome, @Seeit !

As post #1 has been edited to answer @BvU questions, I suggest considering two things:

-The inverse proportionality between p and V.
-The similarity between polygons 1CBA and 1234 due to the axis connecting 1, B and 3, which makes their corresponding sides proportional.

View attachment 324057
So am I right about thinking that A and C are on the same isotherm and have therefore the same temperature?
 
  • #5
Seeit said:
So am I right about thinking that A and C are on the same isotherm and have therefore the same temperature?
Correct!
 
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FAQ: Calculate these 5 temperatures along this Thermodynamic cycle

What are the initial conditions of the thermodynamic cycle?

The initial conditions typically include the initial temperature, pressure, and volume of the working substance. These parameters are crucial for calculating other state variables throughout the cycle.

How do I determine the temperatures at each state point?

The temperatures at each state point can be determined using the ideal gas law and the specific process equations (isothermal, adiabatic, isobaric, or isochoric) that govern the transitions between states. For example, during an isothermal process, the temperature remains constant, while during an adiabatic process, the temperature can be found using the relation \( T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1} \) for an ideal gas.

What equations are needed for an isothermal process?

For an isothermal process, the temperature remains constant. The key equation used is the ideal gas law, \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. Since \( T \) is constant, any changes in \( P \) and \( V \) are inversely proportional.

How do I calculate the temperature change during an adiabatic process?

In an adiabatic process, no heat is exchanged with the surroundings. The temperature change can be found using the adiabatic relation for an ideal gas: \( T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1} \) or \( T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{(\gamma-1)/\gamma} \), where \( \gamma \) is the heat capacity ratio \( C_p/C_v \).

What are common mistakes to avoid when calculating temperatures in a thermodynamic cycle?

Common mistakes include not correctly identifying the type of thermodynamic process (isothermal, adiabatic, isobaric, or isochoric), ignoring the specific heat capacities for non-ideal gases, misapplying the ideal gas law under non-ideal conditions, and neglecting the conservation of energy principles. Ensuring accurate initial conditions and process identification is critical for correct temperature calculations.

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