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Introduction to Thermodynamics

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Category: Introduction to thermodynamics
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Thermodynamics studies heat, work, energy, and the changes they produce in the states of systems. Thus, a system changes its thermodynamic state by exchanging heat or work with other systems with which it interacts.
We will study equilibrium thermodynamics, systems start from a state of equilibrium and through reversible processes (which take place through infinitesimal changes) reach new states, also of equilibrium.

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Category: Introduction to thermodynamics
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We call the part of the universe under study a system . The part of the universe that can interact with the system is called the surroundings .
 

Read more: Thermodynamic systems

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Category: Introduction to thermodynamics
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The walls of a system can be classified as: rigid or mobile , permeable or impermeable, conductive or adiabatic . The mobile walls are deformed allowing the change of the volume of the system. Pervious walls allow the passage of matter through the wall. The conductive walls allow the passage of heat, while the adiabatic ones do not (they are equivalent to the walls of a thermos).

Read more: Walls in Thermodynamic Systems

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Category: Introduction to thermodynamics
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An isolated system is in equilibrium when its macroscopic properties (pressure, volume, temperature) do not change with time. In a non-isolated system two conditions must be met: that the properties of the system do not change with time and that when the system is isolated from the surroundings it does not suffer any variation in its thermodynamic properties.
 
A system is in thermodynamic equilibrium when the following types of equilibrium are fulfilled:
Mechanical equilibrium: All parts of the system are at the same pressure and this coincides with that of the surroundings. Both the external and internal forces acting on the system are balanced.
Thermal equilibrium: The entire system and the surroundings are at the same temperature.
Material Equilibrium: Chemical reactions do not exist or have reached equilibrium and there is no net flow of matter from one part of the system to another or between the system and the surroundings.

 

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Category: Introduction to thermodynamics
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Thermodynamics characterizes a state of equilibrium by properties such as volume, pressure, temperature, composition.
Thermodynamic properties can be classified as intensive and extensive . Those that do not depend on the amount of matter in the system (pressure, temperature, composition) are intensive. The extensive ones depend on the size of the system (mass, volume).
When a thermodynamic system is in a certain state of equilibrium, each thermodynamic variable will take on a specific value characteristic of that state of equilibrium. If we make the system evolve towards another state of equilibrium, the thermodynamic variables will be modified. Thermodynamic variables are also called state functions, since their value depends only on the equilibrium state and not on the path followed to reach it. For example, the thermodynamic variables of a system formed by water at 25ºC are the same whether you prepare it by melting ice or condensing water vapor.

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Category: Introduction to thermodynamics
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A system is homogeneous if any intensive property remains constant throughout the entire system (the same density and composition at every point in the system).

When a system is not homogeneous it consists of a series of parts with different properties called phases. An example of a heterogeneous system is the solution of NaCl in water that is in equilibrium with solid NaCl. This system has two phases, the aqueous and the solid.

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Category: Introduction to thermodynamics
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Just as two systems in mechanical equilibrium have the same pressure, two systems in thermal equilibrium have the same temperature.
Consider two systems A and B in thermal equilibrium with each other (separated by a thermally conductive wall). Similarly, let us assume that B and C are also in thermal equilibrium. It is logical that A and B are at the same temperature and also B and C, but what is the relationship between the temperature of A and C? It is observed experimentally when putting systems A and C in contact through a thermally conductive wall that no heat flow occurs, that is, A and C are also at the same temperature.
 
This experimental fact is known as the zero principle of thermodynamics: “Two systems are in thermal equilibrium with a third one with each other”.

 

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Category: Introduction to thermodynamics
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An ideal gas is a set of atoms or molecules that move freely without interactions. The pressure exerted by the gas is due to the collisions of the molecules with the walls of the container. The ideal gas behavior is at low pressures, that is, in the limit of zero density. At high pressures the molecules interact and intermolecular forces cause the gas to deviate from ideality.

 

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Category: Introduction to thermodynamics
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Boyle in 1662 found a relationship between the pressure of a gas and the volume it occupies. For a fixed amount of gas at constant temperature, he observed that the product of pressure and volume remained constant:

If we graphically represent the product PV against P, it is observed that Boyle's law only holds at low pressures (the curve is horizontal). At high pressures the deviation with respect to Boyle are very important.
Therefore, Boyle's law holds for gases that behave ideally (low pressure conditions).

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Category: Introduction to thermodynamics
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Charles in 1787 and Gay-Lussac in 1802 studied the thermal expansion of gases and found a proportionality between volume and temperature, called Charles Gay-Lussac's law.


For a given mass of gas and constant pressure it is true that the ratio of the volume occupied by a gas and the temperature at which it is found is constant. This law gives fairly accurate results under low pressure conditions.

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Category: Introduction to thermodynamics
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Boyle's law applies to processes that keep the gas mass and temperature constant, on the other hand, Charles' law keeps the gas mass and pressure constant. Let us now consider a process in which pressure, volume and temperature change.


We are going to divide the process into two stages, in the first the temperature is kept constant and in the second the pressure.
Let's apply Boyle's law to the first stage:



Now we will apply Charles's law to the second stage:



Equating both equations:



A m = constant, where m is the mass of the gas, it can be written:



Let us now consider the variation of the mass, keeping the pressure and temperature of the gas constant, we arrive at the relationship: , since the mass is directly related to the number of moles, the previous equation can be written as: , where n represents the number of moles of the gas.

The constant of proportionality k, is a universal constant, called the ideal gas constant and it is customary to represent it by R. Thus, the equation of state of ideal gases can be written as:


Where R = 0.082 atml/Kmol = 8.314 J/molK = 1.987 cal/molK
This law is only exact in the limit of zero pressure.

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Category: Introduction to thermodynamics
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In 1810 Dalton established that the total pressure of a mixture of gases is equal to the sum of the pressures that each gas would exert if it were alone in the container.
Dalton's Law: "The sum of the partial pressures of each gas is equal to the total pressure of the gas mixture."

Let be a mixture of $n_1, n_2....$ moles of gases. The total pressure can be calculated with the equation:



Another way of expressing Dalton's law, very useful in solving problems is: $P_i =x_iP$ , where $P_i$ is the partial pressure of gas i, $x_i$ is the mole fraction of gas i in the mixture of gases and P is the total pressure.


 

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Category: Introduction to thermodynamics
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The relationship between the state variables of a system (P, V, T, n) is given by an equation called the equation of state.
As an example we have the ideal gas equation of state PV=nRT. Only an ideal gas fulfills this equation of state and, as can be seen, it relates the state variables, so that once three of them are known, the fourth can be obtained.

We can ask ourselves what happens with real gases, are there equations of state for them? The answer is yes, one of the best known equations applicable to real gases is the van der Waals equation which has the following form:



a and b are characteristic constants of each gas. If a and b are zero, the ideal gas equation is obtained. The term nb that subtracts from the volume takes into account the molecular repulsion. The term an 2 /V 2 takes into account the molecular attraction that tends to cause the pressure to be less than that predicted by the ideal gas equation.

There are six very useful partial derivatives in thermodynamics, because they can be measured very easily:

These six relations are not independent since three of them are the inverses of the other three.



In addition, we can propose one more equation that relates three of the partial derivatives:

This relationship allows us to clear one of the relationships based on the other two. Therefore only two of the partial derivatives are independent and the remaining four can be calculated from them.
These independent derivatives are known as thermal expansion or coefficient of cubic expansion (\(\alpha\)) and isothermal compressibility (\(\kappa\)).

The relationship between both coefficients is given by the expression:




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Category: Introduction to thermodynamics
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In thermodynamics it is necessary to use some mathematical tools such as the derivative. In this video I explain the definition of derivative and apply it to the function $y=x^2$. In the final part we define the partial derivative of a function.

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Category: Introduction to thermodynamics
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In the following video, the derivative of a composite function is found by applying the chain rule. The total differential of a function of two variables is calculated and some useful relationships are obtained from it, such as the inverse derivative and the -1 rule.