Interstellar space, far from any stars, is filled with a very ...

Question

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H₂). The number density is about 1 atom/cm³ and the temperature is about 3 K. What is the edge length L of an L ✕ L ✕ L cube of gas with 1.0 J of thermal energy?

Accepted Answer

Hello, fellow physicist today we to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Suppose a region of space has interstellar mono atomic gas helium atoms only fictitious though at a density of one atom in every 1000 millimeters cubed and T equals two. Kelvin find the radius R of a spherical volume of the gas that has 5.0 joules of heat energy. So that's our end goal. So ultimately, we're trying to figure out what this value for the radius cap, which is denoted as capital R for the spherical volume of the gas that has 5.0 joules of heat energy. Awesome. So that's what we're ultimately trying to solve for. If we're trying to figure out what R is awesome. We're also given some multiple choice answers. They're all in the same units of meters. So let's read them off to see what our final answer might be. A is three multiplied by 10 to the power of five B is four, multiplied by the 10 to the power of five C is three multiplied by 10 to the power of seven and D is four, multiplied by 10 to the power of seven. OK. So first off, let us recall and use the equation for thermal energy. And let's call this equation one. So let's denote the thermal energy as E subscript, th is equal to three halves multiplied by capital N multiplied by K subscript, capital B multiplied by capital T. And let's note that N is not given or provided in the prom itself. But we are told that in the problem, we're told that NN divided by V is equal to one atom per 1000 millimeters cubed multiplied by 1000 millimeters in 1 m to the power of three is equal to 10 to the power of six atoms per meters. H Thus, we can write that N is equal to 10 to the power of six Adams per meters cubed multiplied by capital V where capital V is the volume. And let's also note and let's write this in blue that K subscript B is Boltzmann's constant, which is 1.38 multiplied by 10 to the power of negative 23 and its units are joules per Kelvin. And we also know that our temperature is two Kelvin. Thus, when we recall the volume of a sphere. So, oh yeah, before we talk about the volume of sphere, let's quickly. So we don't forget that our thermal energy is equal to 5.0 joules. So now we need to recall the volume of a sphere. So let's denote it as V subscript S is equal to four thirds multiplied by pi multiplied by radius cubed. Therefore, we can rewrite equation one as the thermal energy is equal to three halves multiplied by 10 to the power of six atoms per meters cubed multiplied by the volume multiplied by KB multiplied by T. But we can plug in the volume of a sphere in for V. So let me do that. The thermal energy will be equal to three halves multiplied by 10 to the power of sixth atoms per meters cubed multiplied by four thirds multiplied by pi multiplied by R cubed multiplied by KB multiplied by T which we can simplify and write that it's equal to two multiplied by 10 to the power of six atoms per meters cubed multiplied by pi multiplied by radius cubed or RQ multiplied by KB multiplied by T. So now we need to rearrange this equation to isolate and sulfur R the radius, which is our final answer, which I've been writing R as a lower case letter. But in the problem itself, it's written as a capital R but it's the same thing. So with that in mind, when we isolate and sulfur R by itself, you will get that R cubed is equal to the thermal energy divided by two multiplied by 10 to the power of six atoms per meters cubed multiplied by pi multiplied by KB multiplied by T which is equal to the thermal energy which is 5.0 joules. So two and then so 5.0 joules divided by two multiplied by 10, the power of six atoms. And we're plugging in our known variables right now divide so atoms per meters cubed multiplied by pi multiplied by one point 38 multiplied by 10 to the power of negative 23 jewels per Calvin multiplied by two Calvin. Awesome. So now let me plug that into a calculator. We will get that R cubed is equal to 2.88 32 multiplied by 10 to the power of 16 meters cubed. But we need to get our all by itself. So we need to take the cube root. So R is equal to the cube root of 2.8 832 multiplied by 10 to the power of 16 m cubed. Awesome. And when we do that, we will find that R which is equal to capital R is equal to when we round to the nearest hole number or to one significant digit, it will be three multiplied by 10 to the power of 5 m or it's the same as saying 300 kilometers. And this is our final answer. Hooray, we did it. So let's quickly as a side note here, let's note that the low density of atoms compared to 10 to the power of 19 atoms per centimeters cubed in the air that we breathe. And to some extent, a low temperature of two Kelvin means that a large volume is required for 5.0 joules of energy. Note that since the molar mass is not involved, the type of gas used for this particular experiment is not important. Awesome. So with that in mind, let's look at our multiple choice answers to see which of these matches the answer that we found together. And it corresponds with multiple choice answer letter A which is three multiplied by 10 to the power of 5 m. Thank you so much for watching. Hopefully, that helped and I can't wait to see in the next video. Bye.

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H₂). The number density is about 1 atom/cm³ and the temperature is about 3 K. What is the edge length L of an L ✕ L ✕ L cube of gas with 1.0 J of thermal energy?

Key Concepts

Thermal Energy

Ideal Gas Law

Volume and Density

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