16 February 2012

The isothermal process

Vai alla versione in italiano pubblicata il 30 Dicembre 2011

A gas performs an isothermal process when it does or undergoes some volume work receiving or dissipating heat in order to maintain the same temperature.
When a gas expands it does volume work, the volume increases while pressure and temperature decrease.
Providing heat from the outside the temperature change can be avoided, but the pressure drops, even if a minor change occurs compared to the case where the temperature is not kept constant.
When a gas is compressed thus undergoing volume work, the volume decreases while pressure and temperature increase.
Dissipating heat to the outside the temperature change can be avoided, but the pressure grows, even if a minor change occurs compared to the case where the temperature is not kept constant.
In practice, an isothermal change with heating is an expansion at a constant temperature and at the same time a depressurization at a constant temperature; a transformation isotherm with loss of heat is a compression at a constant temperature and at the same time a pressurization at constant temperature.

At the same heat exchanged, the volume change and pressure does not depend on the gas type, but they depend only on the initial values ​​of P and V and on the amount of gas according to the following relations

Volume change = Vfinal - Vinitial = Vinitial * { e [ Q / ( n * R * T ) ] - 1 }

Pressure change = Pfinal - Pinitial = Pinitial * { e [ - Q / ( n * R * T ) ] - 1 }

where

Vfinal is the volume at the end of the process expressed in m3
Vinitial is the volume at the beginning of the process expressed in m3
Pfinal is the pressure at the end of the process expressed in Pa
Pinitial is the pressure at the beginning of the process expressed in Pa
Q is the exchanged heat expressed in J: a positive value of Q means heat supplied to the gas, a negative value of Q means heat lost by the gas
n is the gas quantity expressed in mol
R is the ideal gas constant (R = 8,314 J mol-1 K-1)
T is the temperature of the process expressed in K

Below the volume and pressure changes are plotted as function of the exchanged heat.
The values ​​refer to 1m3 of gas at T=300K, P=100kPa, n=40.09 mol.

The points where the exchanged heat is positive are relative to heat supplied to the gas, the points where the exchanged heat is negative are relative to the heat lost by the gas.
The first graph highlights the fact that the volume does not change linearly with the heat exchanged, but instead it has a exponential trend (of the type y=ex).
The second graph shows the fact that the pressure does not change linearly with the heat exchanged, but instead it has a trend of the reverse exponential type (of the type y=e-x).
The graphs presented below show the actual values ​​of the volume and pressure.

The first graph highlights once again the fact that the volume does not vary linearly with the heat exchanged, but instead, it has a trend of exponential type
(general form y=ex+Constant).
The second graph shows the fact that the pressure does not vary linearly with the heat exchanged, but instead, it has a trend of reverse exponential type (general form y=e-x+Constant).

The isothermal process in a PV diagram is represented by a curve of hyperbolic type.
Starting from the ideal gas state equation

P * V = n * R * T

we arrive immediately at

P = n * R * T / V

where the term n * R * T is constant for isothermal processes then the equation takes the simplified form

P = constant / V

Starting from the same initial state, that is at the same P, V, n and T, and exchanging the same heat, the length of the curve is gas type independent.
In the following picture two processes have plotted.
The one that develops from the initial point to the left is an isothermal compression with 100kJ of heat loss. The other is an isothermal expansion with 100kJ of supplied heat.

A key consideration is that in the isothermal process there is the volume work (with positive sign if the gas does the work, with negative sign if the volume work has done on the gas). Its value in the PV diagram is the area under the transformation curve (the red area for the isothermal compression, the green area for the isothermal expansion).

Mathematically, the area subtended by a curve is calculated by integration and in the specific case is written as follows

Lisothermal expansion = n * R * T * ln ( Vfinal / Vinitial ) = 40.09 mol * 8.314 J mol-1 K-1 * 300 K * ln ( 2.718 m 3/1 m 3 ) = 100kJ = Q

Lisothermal compression = n * R * T * ln ( Vfinal / Vinitial ) = 40.09 mol * 8.314 J mol-1 K-1 * 300 K * ln ( 0.368 m 3/1 m 3 ) = -100kJ = Q

In an isothermal expansion, the heat supplied is fully converted into work done by the gas.
In an isothermal compression the work done on the gas is converted entirely into dissipated heat.
For this reason the thermal energy of a gas is constant during an isothermal process.
The isotherm is a process in which the only forms of involved energy are the heat exchanged and the work.
In the isothermal expansion (or even isothermal depressurization), the energy supplied to the gas in the form of heat is completely converted into volume work.
In the isothermal compression (or even isothermal pressurization), the gas receives energy as work from the outside and the gas fully dissipates this energy as heat.
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8939334750175476, "perplexity": 702.1192847863447}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637900397.29/warc/CC-MAIN-20141030025820-00237-ip-10-16-133-185.ec2.internal.warc.gz"}