Calculate the speed of sound at 20^∘ C for (a) hydrogen, (b) helium, (c) methane, (d) nitrogen,

Calculate the speed of sound at 20^∘ C for (a) hydrogen, (b)...

Key Concepts

Temperature Conversion

Temperature plays a vital role in determining the speed of sound in a gas, and the absolute temperature (in Kelvin) must be used in the calculations. Converting from degrees Celsius to Kelvin ensures that the energy scales in the formula are consistent with the thermodynamic principles underlying the speed of sound, emphasizing the importance of proper unit usage in physical formulas.

Universal Gas Constant (R)

The universal gas constant, R, is a fundamental constant in physics and chemistry that links temperature, energy, and the amount of substance in the ideal gas law. In the context of sound propagation, it helps in expressing the relationship between temperature and molecular motion, thereby influencing the speed at which sound waves travel through a gas.

Molar Mass

Molar mass (M) is a critical property of a gas as it influences the inertia of the gas molecules. In the expression for the speed of sound, a lower molar mass leads to a higher speed because lighter molecules can respond more quickly to pressure changes. When calculating the speed of sound, the molar mass of each gas must be appropriately considered to account for differences in molecular weights.

Adiabatic Index (?)

The adiabatic index, denoted as ?, is the ratio of specific heats at constant pressure (Cp) and constant volume (Cv). It represents the stiffness of the gas under adiabatic (no heat transfer) conditions, affecting how pressure and density change as a sound wave travels through the medium. A higher ? indicates that the gas is less compressible, leading to faster sound propagation.

Speed of Sound in Gases

The speed of sound in gases is determined by how quickly pressure variations (sound waves) propagate through the medium. It depends on the gas’s thermodynamic properties, and for an ideal gas is typically calculated using the formula c = ?(?RT/M), where c is the speed of sound, ? is the adiabatic index, R is the universal gas constant, T is the absolute temperature, and M is the molar mass. This formula encapsulates how molecular weight and thermal properties influence the propagation speed of sound.

Transcript

00:01 So here in this problem we have to calculate the speed of sound at 20 degrees celsius temperature is given as 20 degrees celsius for five different gases so for the part a we have to calculate for hydrogen so this the basic equation to calculate the speed of sound is given by c is equals to under root k rt so first we'll be calculating the speed for hydrogen so it will be equals to under root k for hydrogen it is 1 .41 times r for hydrogen it is 4 124 jule over k g kelvin times t it is given as 20 degrees celsius but we have to convert this thing into kelvin so we'll get 293 kelvin we're going to simplify this and we'll get the value as 1 ,305 meter per second now next we have to solve it for methane sorry for helium for the b part we have to solve for helium so we'll use the same formula so speed of sound for helium in case of helium will be equals to under root k for helium it is 1 .66 times k r so r for helium is 2077 jule over k g kelvin times t is 293 kelvin so we'll get the value as 1 ,0005 median per second now next for the c part we have to solve it for methane that is ch4 so we'll be using the same formula that is c for ch4 will be equals to under root k for methane is 1 .31 times r for methane is 518 .3 jule over kg kelvin times t is 293 kelvin so we're going to simplify this and we'll get the value of ch4 as 446 meter per second now in the d part we have to solid for nitrogen so again we'll be using the same formula so c4 n2 will be equals to under root the value of a k for a nitrogen it is 1 .40 times r is 296 point 8 times t is 293 kelvin so we'll get the value as 349 meter per second now next we have to solid for in the d part sorry in the e part you have to solve it for co2 so same using the same formula c for co2 will be equals to under root k for co2 is 1 .29 times r is one one one eighty eight point nine and t is same that is 293 kelvin so we'll get the value as for c co2 as 290 267 meter per second so these are the results for different gases this one this one this is for ch4 this is for n2 and this is for co2 and this is for co2 so i hope you understood the problem...

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