# TDS ratio specific heat help

• foranlogan2
In summary, based on the definitions of these terms, it can be said that the first equation is calculating the heat capacity (or thermal energy) of a substance as a ratio to another substance. The second equation is calculating the entropy of a substance as a ratio to another substance.
foranlogan2
Explaining your reasoning, show that the ratio of the specifc heat is

CP/CV = (DP/DV)s / (DP/DV)t

hint :use TDS EQUATIONS

I have used tds equation 3 and tds equation 1 and equated them,but i am stuck because i get to an equation like

(CP/CV-1)(DT/DV)p = T(DP/DT)v and just do not know how to get to the ratio

that is same as to right in the expression below:
$$\gamma$$ = ks/ kt
which $$\gamma$$ is the ratio of CP and CV heat capacities and ks and kt are the isentropic and isothermal compressibility, respectively.
k can be expressed as below:
kt= -(1/V)(DV/DP)t
and,
ks= -(1/V)(DV/DP)s

once we know these, all you should worry about is to have the ratio of heat capacities ( CP and CV ) out of Tds Equations, and that is by using the first and second ones and the third one is of no use here, there we have:

Tds= CVdT + T(DP/DT)VdV (the first Tds equation)
and,

Tds= CPdT - T(DV/DT)PdP (the second Tds equation)
by putting dS=0 in both equation we wil have,

CV = - [T(DP/DT)VdV]/dT
and,

CP = [T(DV/DT)PdP]/dT
by dividing the two new equations we will have,

CP/CV = - [(DV/DT)PdP]/[(DP/DT)VdV]
more explicitly,

CP/CV = - [(DV/DT)P/(DP/DT)V](DP/DV)?
we should put "S" instead of "?" for started with the assumption ds=0 and that is how got here. That is,

CP/CV = - [(DV/DT)P/(DP/DT)V](DP/DV)S
next time we put dT=0 in the Tds equations (the first and second), then we should have,

ds= (DP/DT)VdV
and,

ds= - (DV/DT)PdP
left sides are equal, so we try putting right sides equal,

(DP/DT)VdV = - (DV/DT)PdP
more explicitly,

(DP/DT)V/(DV/DT)P= - (DP/DV)?
this time we should put "T" instead of "?" for we started with the assumption dT=0. That is,

[(DP/DT)V/(DV/DT)P]= - (DP/DV)T
with a small change we will have,

[(DV/DT)P / (DP/DT)V] = - [ 1/ (DV/DP)T]
from there we will have:

CP/CV = - [- 1 / (DV/DP)T](DP/DV)S
no different from,

CP/CV = (DP/DV)S / (DV/DP)T
================================
one question students may get up to, after working out this problem, is:
Why using TdS equations to solve the problem?
and the answer is not that simple!
you may or maynot find it out later.

One may say it is of the two equal interpretations on thermodynamics, which say:
1- thermodynamics is the science dealing with work, heat and internal energy
2- thermodynamics is the science dealing with entropy and energy.

The TDS equation 3 and TDS equation 1 are not the correct equations to use in order to show the ratio of specific heat. Instead, we need to use the relationship between internal energy (U), enthalpy (H), and the first and second laws of thermodynamics:

dU = TdS - PdV

dH = TdS + VdP

By rearranging these equations and dividing the second by the first, we get:

dT/dV = (dP/dV)s / (dP/dV)t

We can then use the definition of specific heat (CP/CV = (dT/dV)s) to substitute in for the left side of the equation:

CP/CV = (dP/dV)s / (dP/dV)t

And since the partial derivatives (dP/dV)s and (dP/dV)t are equal to the ratio of specific heats (CP/CV), we can substitute them in to get the final ratio:

CP/CV = (CP/CV)s / (CP/CV)t

Therefore, the ratio of specific heat is equal to the ratio of the specific heats at constant pressure and constant volume. This relationship holds true for any substance, and can be used to calculate the specific heat at constant pressure or volume if the other is known.

## 1. What is the TDS ratio specific heat?

The TDS ratio specific heat is a measure of the amount of heat required to raise the temperature of a substance by a certain amount, taking into account the total dissolved solids (TDS) present in the substance.

## 2. How is the TDS ratio specific heat calculated?

The TDS ratio specific heat is calculated by dividing the specific heat by the TDS ratio. The specific heat is a physical property of a substance, while the TDS ratio is the ratio of the total dissolved solids to the total mass of the substance.

## 3. Why is the TDS ratio specific heat important?

The TDS ratio specific heat is important because it allows scientists and engineers to accurately predict how much heat is needed to raise the temperature of a substance, taking into account the presence of dissolved solids. This is crucial in various industries such as food and beverage, chemical, and environmental engineering.

## 4. How does the TDS ratio affect specific heat?

The TDS ratio affects specific heat because dissolved solids in a substance have their own specific heat values, which can change the overall specific heat of the substance. This means that substances with higher TDS ratios will require more heat to raise their temperature compared to substances with lower TDS ratios.

## 5. Can the TDS ratio specific heat vary for the same substance?

Yes, the TDS ratio specific heat can vary for the same substance depending on the concentration of dissolved solids present. This is because the TDS ratio is directly proportional to the specific heat, meaning that as the TDS ratio increases, the specific heat also increases.

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