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Material Properties
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The material constants, whether for gases, liquids, or solids, typically represent the relationships between various variables. In this context, material constants correspond to slopes in different combinations of variables.
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Properties of Materials
Concept
Material properties generally describe how various variables change among them. The primary variables that characterize the state of a gas, liquid, and solid are:
• the pressure ($p$)
• the absolute temperature ($T$)
• the volume ($V$)
• the entropy ($S$)
The first two are intensive variables, meaning they do not depend on the size of the system. Therefore, any variation will simply be equal to:
• the pressure Variation ($dp$)
• the temperature variation ($dT$)
In the case of extensive variables, there is a dependence on the size of the system. Therefore, in this case, the variable must be normalized by dividing it by the system's size:
• the volume Variation ($dV$) divided by the volume ($V$)
• the entropy variation ($dS$) divided by the entropy ($S$)
Since the number of variables is fixed, there are only a limited number of alternatives and, consequently, constants.
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Model
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Equations
$ c ^2=\left(\displaystyle\frac{ \partial p }{ \partial \rho }\right)_ S $
c ^2= dp / drho
$ C_p - C_V = n R $
C_p - C_V = n R
$ C_p = T DS_{T,p} $
C_p = T * @DIFF( S , T , 1 , p )
$ C_V = T DS_{T,V} $
C_V = T * @DIFF( S , T )
$ k_p =-\displaystyle\frac{ DV_{p,T} }{ V }$
k_p =- DV_pT / V
$ k_T =\displaystyle\frac{ DV_{T,p} }{ V }$
k_T = DV_Tp / V
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Constant pressure heat capacity
Equation
Specific heat capacity is defined as the change in temperature with respect to the supplied or extracted heat. It can be expressed by the equation:
$\delta Q = C_p dT = T dS$
This equation is an inexact differential because it depends on how the heat is supplied or extracted. In particular, when considering a constant pressure process, we define the heat capacity at constant pressure.
In other words:
| $ C_p = T DS_{T,p} $ |
where $C_p$ is the heat capacity at constant pressure.
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Constant volume heat capacity
Equation
The heat capacity is defined as the change in temperature with respect to the supplied or removed heat. It can be expressed using the equation:
$\delta Q = C dT = T dS$
This equation represents an inexact differential, as it depends on the manner in which the heat is supplied or removed. In particular, when considering a process carried out at constant volume, we define the heat capacity at constant pressure.
In other words:
| $ C_V = T DS_{T,V} $ |
Here, $C_V$ represents the heat capacity at constant volume.
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Mayer's ratio for caloric capacity
Equation
The Mayer's relation states that the heat capacities of a gas at constant pressure and constant volume are related by the universal gas constant and the number of moles, as expressed by:
| $ C_p - C_V = n R $ |
Here, $C_P$ represents the heat capacity at constant pressure, $C_V$ represents the heat capacity at constant volume, $n$ is the number of moles, and $R$ is the universal gas constant.
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Isothermal compressibility coefficient
Equation
Compression is defined using as
| $ \kappa \equiv-\displaystyle\frac{1}{ V }\left(\displaystyle\frac{\partial V }{\partial p }\right)_ T $ |
When the notation is employed, the compressibility coefficient is defined as
| $ DV_{p,T} \equiv\left(\displaystyle\frac{ \partial V }{ \partial p }\right)_ T $ |
The compressibility coefficient itself is defined through as
| $ k_p =-\displaystyle\frac{ DV_{p,T} }{ V }$ |
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Coefficient of thermal expansion
Equation
Thermal expansion is defined using as
| $ k_T \equiv \displaystyle\frac{1}{ V } \left(\displaystyle\frac{\partial V }{\partial T }\right)_ p $ |
When the notation is employed, the coefficient of thermal expansion is defined as
| $ DV_{T,p} \equiv\left(\displaystyle\frac{ \partial V }{ \partial T }\right)_ p $ |
The coefficient of thermal expansion itself is defined through as
| $ k_T =\displaystyle\frac{ DV_{T,p} }{ V }$ |
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Sound velocity as a derivative of the pressure
Equation
Sound is an oscillation of density that propagates and is associated with a corresponding variation in pressure. Therefore, the speed of sound squared ($m^2/s^2$) can be defined as the ratio of the pressure variation ($Pa = kg/m s^2$) to the density ($kg/m^3$). Due to the short time in which this occurs, it is assumed to be a variation at constant entropy. Thus, we can express it using as follows:
| $ c ^2=\left(\displaystyle\frac{ \partial p }{ \partial \rho }\right)_ S $ |
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