![]() Similarly, all of the springs have the same stiffness $k$ and length $h$. The balls all have the same mass which we denote by $m$. Our model consists of a series of balls connected by springs as in the diagram below: In this section we will study a simpler model of compression waves and extend our results to sound waves later. Sounds waves are an example of a compression wave. Using the ideal gas law and the first law of thermodynamics we’ll see that the “stiffness” of such a spring is determined by the temperature and heat capacity of the gas.įinally, we’ll put everything together to write down a formula for the speed of sound in a gas in terms of its mass, temperature and heat capacity and verify that it is consistent with the table. We will then show that a pneumatic tube filled with gas behaves quite similarly to a spring. The key result will be a formula for the speed of such a wave in terms of the mass of the balls and the stiffness of the springs. In the next section we study what happens when a wave passes through a series of balls connected by springs. So molecules with more atoms have more opportunities to absorb heat. The general idea is that molecules can absorb energy into the vibrations and rotations of their constituent atoms relative to one another. As we will explain later in more detail, heat capacity is essentially a proxy for the number of atoms. We’ll then test our model on the table above and find that it is surprisingly accurate!īased on the experimental success of our model we’ll conclude that the speed of sound in an ideal gas depends on three factors: temperature, mass and heat capacity. To make sense of these data, we will develop a simple model describing sound waves and use it to derive a formula for the speed of sound using Newton’s laws of motion and basic thermodynamics. ![]() For example, Methane has more atoms than Argon but the speed of sound through Methane is greater. So perhaps the number of atoms is what determines the speed of sound, where a greater number of atoms implies a lower speed? For example, a molecule of Ethane has 8 atoms whereas the noble Argon has only one. How can we make sense of these outliers? Note that in both cases the lighter molecule nevertheless contains more atoms. For example, Argon is heavier than Ethane but sound travels more quickly through Argon. But clearly mass is not the only relevant factor. First of all, the speed of sound seems to decrease as the mass of the gas increases. NameĪfter staring at this table a bit a few trends emerge. These speeds are for gases at standard atmospheric pressure (1 atm) and the indicated reference temperatures. ![]() To make sure we are grounded in reality, we would specifically like to explain this table of the speed of sound through various gases from the Handbook of Chemistry and Physics 2. The goal of this post is to understand how the molecular composition of a gas determines the speed of sound waves passing through it. Perhaps the most popular demonstration of this phenomenon are the modulated voices of people who have inhaled Helium 1. Hence the temperature of the medium does not remain constant.It is well known that sound waves travel through different gases at different speeds. To account for this difference, Laplace pointed out that the compression and rarefactions occur so rapidly that heat of compression remains confined to the region where it is generated and does not have time to flow to the neighboring cooler regions which have undergone an expansion. On substituting the values of atmospheric pressure and density of air at S.T.P in equation ….relation, we find that the speed of sound waves in the air comes out to be 280 ms -1, whereas its experimental value is 332 ms-1. We get Newton’s formula for the speed of sound in air. So, substitute P for E in relation given below: The expression (ΔP/ΔV/V) is the elastic modulus E at a constant temperature. ![]() The product of ΔPΔV is very small and can be neglected. For the calculation of elastic modulus for air, Newton assumed that when a sound wave travels through air, the temperature of the air during compression remains constant and pressure changes from p to (p + Δp), and therefore, the volume changes from V to (V – ΔV). In general, sound travels more slowly in gases than in solids because gases are more compressible and hence have a smaller elastic modulus. This makes sense because the molecules in a solid are closer than in gas and hence, respond more quickly to a disturbance. As seen from the table, the speed of sound is much higher in solids than in gases.
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