2.4. Compression and Expansion of Gases

So, one of the phenomenon’s is isothermal, which is constant temperature. So, what happens is PV = nRT, that equation we already know. So, PV = nRT. So =RT which is constant. Okay? So, this is RT. So, or inversely proportional to p where p is absolute pressure and V is specific volume. And therefore, we can also write So, we need to find out for to be able to find out because is given as . So, . Right? So is and when substituted, it will give us =P. So, more importantly for isothermal process what you must remember is that =P. This is most important finding of this particular slide.

An isothermal process refers to a physical process that occurs at constant temperature. In the context of gases, it means that while the pressure (P) and volume (V) of a gas may change, the temperature of the gas remains unchanged. According to the ideal gas law (PV = nRT), if the temperature (T) is constant, then the product of pressure and volume (PV) must also remain constant. This implies that if you increase the volume (V) of the gas, the pressure (P) must decrease, and vice versa. The equation also suggests that the specific volume (volume per unit mass) increases as pressure decreases under isothermal conditions, highlighting the relationship between the two variables.

Imagine a balloon that you are gradually inflating. As you blow air into the balloon, it expands, increasing its volume (V). If you’re blowing up this balloon slowly enough to keep its temperature constant, the pressure inside the balloon (P) will decrease. It’s like trying to fill a large flexible balloon with air—it expands (volume increases), and to keep everything balanced, the pressure inside must drop.

So, we are going to look at another phenomenon or process called Isentropic where no heat is exchanged. In this case, the equation for isentropic process is given by where . This is a very standard terminology in thermal physics that you have already seen in your class 10th and 12th. So, this is also called a specific heat ratio. So if you do from this equation, you will get therefore . We are using the substituting the value of C in terms of and therefore, , or we can also write , because, is given as . Therefore, we have got , so we get more importantly in case of Isentropic process is given by , k is specific heat ratio. So for isothermal process, this is going back and for Isentropic process , that is very important to note.

An isentropic process is an idealized process in which there is no heat transfer to or from the system, meaning that the process occurs adiabatically, and it is both reversible. In isentropic processes, the change of state of a gas is characterized by the ratio of specific heats (k), which relates the pressures and volumes at two states of the gas. For example, it can be shown that during an adiabatic process, when a gas expands, the temperature drops; when it is compressed, the temperature rises. The key takeaway for isentropic processes is to remember the relationship between pressure, volume, and temperature under conditions of adiabatic transformation of gases.

Think of a bicycle pump. When you quickly compress the air inside the pump by pressing the handle down, no heat is lost to the environment (adiabatic). The air (gas) inside heats up (because work is done on it), and when you stop, the compressed air stays hot for a while, embodying the principles of an isentropic process.

So another important thing that we should be aware of speed of sound is speed of ‘c’ is given as this is the formula and we know that it is . Now, we have to solve for . We already know from this equation that can be written as. Right? Therefore, if we put this equation in this equation, then we can get ‘c’ as , where ‘c’ is generally very large for compressible, I mean it is very difficult to compress fluids because in case of fluid is too much the bulk modulus. So, we should be if we know this equation and we know what type of process is there we should be easily able to find out the speed of sound.

The speed of sound in a medium is influenced by the medium's properties, particularly its density and bulk modulus. The general formula to determine the speed of sound (c) in a gas is dependent on the equation linking pressure, density, and the specific heat ratio. In gaseous media, because they are more compressible than liquids or solids, understanding their properties can help predict how sound waves travel through them. Essentially, gases can change pressure more easily than liquids or solids, affecting how rapidly sound travels through them.

Consider the difference in sound when you listen to someone talking underwater vs. in air. Sound travels faster in water because water is denser than air; thus, the particles in water are closer together, allowing sound waves to transmit energy more quickly. This makes it easier for sound to travel through liquids, highlighting the principles underlying the speed of sound in various media.

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