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What is the relation between the velocity and temperature? |
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Answer» Solution :Let us consider an ideal gas WHOSE equation of state is PV = nRT …………(1) where, P is PRESSURE, V is volume, T is temperature, n is number of mole and R is universal gas constant. For a given mass of a molecules, equation (1) can be WRITTEN as `(PV)/T`= Constant ...............(2) For a FIXED mass m, density of the gas inversely varies with volume, i.e., `rho propto 1/V, V=m/rho`........(3) Substituting equation (3) in equation (2), we get `P/rho = cT`..........(4) where c is constant. The speed of sound in air given in equation `v_(A) = sqrt((B_(A)/rho)) = sqrt((gamma P)/rho) = sqrt(gamma v_(T))` cna be written as `v= sqrt((gamma P)/rho) = sqrt(gamma cT)`............(5) SINCE `v propto sqrt(T)`, the speed of sound varies directly to the square root of temperature in kelvin. Let `v_(0)` be the speed of sound at temperature at `0^(@)` C or 273 K and v be the speed of sound at any arbitrary temperature T (in kelvin), then `v/v_(0) = sqrt(T/273) = sqrt((273 +t)/273)` `v=v_(0)sqrt(1+t/273) -=v_(0)(1+t/546)` (using binomial expansion). Since, `v_(0) = 331 ms^(-1)` at `0^(@)` C, v at any temperature in `t^(@)C` is `v=(3310.60t)ms^(-1)` Thus, the speed of sound in air increases by `0.61 ms^(-1)` per degree celcius rise in temperature. Note that when the temperature is increased, the molecules will vibrate faster due to gain in thermal energy and hence, speed of sound increases. |
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