What is Volume: Definition and 1000 Discussions

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or 3D shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.
In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.

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  1. Z

    Automotive Loss of gaseous volume because of fuel combustion

    Hi, 1st and general case: 2 H2 and 1 O2 take up 1 liter of volume at, for example, 20°C, before combustion. After combustion and cooling down to 20°C, how much volume do the exhaust gases take up? Since the stoichiometric relationsship is 2 mols of H2 for every mole of O2, so those three moles...
  2. aboredperson

    Chemistry Question about Gas Law Problem

    PV=nRT P*1L = 6 moles * 0.0821*298 (I added up all the moles and solved for pressure) P =146.79 atm 146.79 atm * V = 2 moles Ar * 0.0821* 298 (I plugged in the moles for argon and solved for volume) V= 0.333 L Answer key says the answer is 1 Liter. Where did I go wrong?
  3. Hauzen

    Please tell me the difference in the size of the air volume

    Hi. Since I've been studying fluid mechanics, I've been asking a lot of questions Thank you all for your help! I suddenly have a problem that I want to think about while studying today. I would like to know the change of air volume according to the pipe shape as shown in the picture below...
  4. Hauzen

    I have a question about the air volume

    Hello I need your help studying hydrodynamics. I have a question about the volume of air. Assuming incompressibility, non-viscosity, there is a coefficient of tube friction, and it moves in laminar flow... The Oulet stage Fan is installed, so air moves from top to bottom. The cross-sectional...
  5. S

    B Having trouble deriving the volume of an elliptical ring toroid

    Like the title and the summary suggest, I can derive the volume ##V=2\,\pi^{2}\,r^{2}\,R## for a ring torus - a doughnut-style toroid (one such that the major radius ##R## > the minor radius ##r##, and it therefore has a hole at the center) that is of circular cross-section. But I want to be...
  6. C

    B Question about volume of a sphere

    Hello everyone, I was trying to find volume of a sphere by doing some calculus, the area of a circle is ##{\pi}r^2## So I thought I would calculate the volume of one hemisphere and then multiply by two, but I got a different result than the standard formula, the standard formula is ##\frac 4 3...
  7. J

    I How large is the volume of a photon interacting with matter?

    When a photon interacts with matter and diffracts, the photon seems to interact with the area of its wave front, determined by an aperture even if it is only one photon. But how far does it interact in depth? Does it always "feel" the whole crystal? Can we assume the photon is reflected by the...
  8. B

    How can I calculate the Density and Volume of a mechanical mixture (atoms)?

    Is it correct that: density = [ 0.09 * (density of NaCl) ] + [0.91 * (density of water) ] volume = [ (volume of water) + ( { (volume of NaCl) / 48} * 9) ] Thank you so much for any advice.
  9. D

    Engineering Composition of flue gases by volume on a wet basis and dry basis

    Butane 0.75mol of C4H10 0.75 x C4 = 3mol of CO2 0.75 x H5 = 3.75mol of H2O 0.75 x 6.5O2 = 4.875mol of O2 Propane 0.10mol of C3H8 0.10 x C3 = 0.3mol of CO2 0.10 x H4 = 0.4mol of H2O 0.10 x 5O2 = 0.5mol of O2 Butene 0.15mol of C4H8 0.15 x C4 = 0.6mol of CO2 0.15 x H4 = 0.6mol of H2O 0.15 x...
  10. B

    Volume's effect on buoyancy: Does pressure increase?

    Hey! Im currently writing a lab on how an increase in the volume of an object will result in an increase of buoyancy force acting on an object. We fully immerse different amounts of clay playing blocks into water (using a string) on top of a scale, and calculate the buoyancy force. The reasoning...
  11. chwala

    Find the volume of the solid bound by the three coordinate planes

    Also, $$V=\dfrac{1}{4} \int_4^0 \left[\int_{2-0.5y}^2 (4-2x-y) dx\right] dy$$ $$V=\dfrac{1}{4} \int_4^0\left [4x-x^2-xy]^2_{2-0.5y} \right] dy$$ $$V=\dfrac{1}{4} \int_4^0 \left [(4-2y)-(4(2-0.5y)-(2-0.5y)^2-(2-0.5y)y] \right] dy$$ $$V=\dfrac{1}{4} \int_4^0 \left [(4-2y)-(2-0.5y)^2 \right]...
  12. narrator

    Trying to resolve a volume calculation

    Hi all, I've made errors in my calculations and need help. I'm trying to calculate the volume of a layer above sea level to the height of Flight Level 300 i.e. 30,000 feet. Next, I want to subtract the volume of land within that layer. Here are my calculations. Purely for the sake of my...
  13. S

    Find the volume bounded by hyperboloid and plane z = ± d

    My attempt: The shape of the hyperboloid would be like this: If the hyperbolod is cut by plane z = d, the intersection would be a ellipse. Projecting the intersection to xy - plane, I think I get: $$-2\leq x \leq 2$$ $$-b\sqrt{1-\frac{x^2}{a^2}} \leq y \leq b\sqrt{1-\frac{x^2}{a^2}}$$ So the...
  14. T

    Confusion on product rule for mass of differential volume element

    Good evening, I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context. I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this...
  15. MatinSAR

    Change in volume of sphere with change in temperature

    I am sure that the radius of the two spheres changes equally. But in answer of the question it said that their change in volume is the same. Is it correct?! Link of website: https://www.toppr.com/ask/question/two-spheres-of-same-size-are-made-of-the-same-metal-but-one-is-hollow/ I think it is...
  16. J

    Calculate the bouyancy of two spheres joined by a rope submerged in seawater

    m * g = mAl * g V * ρ * g = VAl * ρAl * g V * ρ * g = V * ρAl * g ρ = ρAl this does not work at all, because the upper ball must have a density smaller than that of seawater 1200kg/m3 or not?
  17. ananonanunes

    Find volume of this object using integrals

    I am given this expression which represents an object in 3D and the goal is to determine its volume using multiple integrals. I started by drawing what I think is the object as well as two "slices" of that object on different planes (z=2 and z=1) I have tried using cartesian, cylindrical and...
  18. A

    Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

    Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
  19. Ebi Rogha

    Gas temperature in a constant volume

    An insulated container (constant volume, adiabatic) contains an Ideal gas with pressure P1 and temperature T1. We open the container's hatch for a few seconds and let some particles escape from the container, then we close the hatch again. We know container's pressure has reduced by exiting...
  20. 1

    MCNP6.2 - Are results of FMESH tallies already divided by cell volume?

    Hi everyone, I am struggling to understand whether the results of FMESH tallies are already divided by the cell volume or not. I'd actually expect so considering: 1. the comparison with an F4 tally in the same cell where results are comparable only if I assume that the mesh tallies results are...
  21. B

    I Time for a cold volume of air to reach a higher air temperature

    I have a cube with a volume of 1000m3 at an initial temp of 290K. The bottom side (10m by 10m) is open to the ambient air. I put this cube into a huge fridge and cool the whole volume by 5K. I close the open side by placing a cover on it. This cube has now got a volume of air at a temperature of...
  22. C

    Solving the Mystery of a Floating Bottle: Density & Volume

    If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above the surface of water. What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant). It's easy problem but I can't get right...
  23. M

    I Adiabatic expansion work far exceeds isobaric of same volume, why?

    Using the adiabatic process formula, I've calculated the change in volume for a volume of gas with an initial pressure of 10 psig expanding to 0 psig. The initial volume is 100 cubic inches and the expanded volume is 144.9. This is a difference of 44.9. The total work done ends up being about...
  24. Pushoam

    Calculating Volume of a Sphere Using Integration: What Mistakes Have I Made?

    I consider a disc of thickness ## R d\theta ## as shown in the figure. Then, $$ dV = \pi R^2 sin^2 \theta R d\theta $$ ( Area of the disc * its thickness) Hence, $$ V = \int^{\pi}_{0} \pi R^2 sin^2 \theta R d\theta $$ $$ V = \frac 1 {2} {\pi}^2 R^3 $$ ....(1) While $$ V =...
  25. E

    I Confusion about the interpretation of specific volume

    So I'm reading through Cengel's thermodynamics textbook, and came across this solved example: Firstly, pressure in this context I'm assuming is vapour pressure? Since we're dealing with pure substances in this chapter. But what's confusing me is, here's the diagram I have: They've not...
  26. A

    B Need some clarification to get dimensions for a volume

    I have a volume of 0.04 cubic centimeters. I need to convert into something I can work with in my shop. Tolerance is not important. To get dimensions of a cube with this volume, I calculated the cube-root which came out to 0.341. So, my cube would be (rounded) 3mm on a side, correct? Next...
  27. tracker890 Source h

    (has been resolved):Integral to find elbow volume

    has been resolved Please help me to understand which answer is correct. I use the integral method to find the elbow volume as follows: But my book say:
  28. P

    I Change of Variables in Double Volume Integral

    In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following. $$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$ $$ \bf z = \bf r^{''} - \bf r^{'} $$ $$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z +...
  29. J

    Specific volume and enthelpy of mercury

    I have been away from thermodynamics for a while, and there is a young mechanical engineering student who asked my help in solving the question. For the life of me I am unable to solve this problem and I am at my wit's end. I believe I can solve the problem if some pointers are provided. Also...
  30. C

    What is the change in internal energy after two processes?

    So the question goes like this: find change in internal energy in process 1->2 using diagram. And offered solutions a)-400J b)400J c)600J d)800J. First I found T1 and T2 using (P*V)/T=R and got T1=24K and T2=72K. Then I found n(number of moles) using PV=nRT and got n1=1mol, n2=1mol. Then I...
  31. S

    I Internal energy of a comoving volume increasing as space expands?

    I was reading an article by Edward Harrison, which tackles the problems of conservation of energy at cosmological scales. At some part (point 2.4) he cites several article, including one by Rees and Gott, which he says indicates that the internal energy of a comoving volume (e.g. a cosmic...
  32. cwill53

    Calculation of Electrostatic Potential Given a Volume Charge Density

    Part (a) was simple, after applying $$Q=\int_{\mathbb{R}^3}^{}\rho \, d^3\mathbf{r}$$ I found that the total charge of the configuration was zero. Part (b) is where the difficulties arise for me. I applied $$V(\mathbf{r})=\frac{1}{4\pi \epsilon _0}\int_{\Gamma }^{}\frac{\rho...
  33. L

    Finding the Volume of a Slightly non-Rectangular Box

    I have a storage tote that has a larger top than bottom. How do I figure out its volume? Is this like a 3D trapezoid? Can I measure the volume of the rectangle assuming the top is the same as the bottom, then as if the bottom were the same as the top, then just subtract the two?
  34. T

    Gas law problem (changing volume, temperature and pressure)

    I can’t quite work my head around this question, I am having a difficult time analyzing the question, I can’t seem to make out what the initial and final conditions are would appreciate all the help I could get cheers
  35. Onyx

    I Appearance of Warp Bubble Internal Volume to Distant Observer

    At a single moment of coordinate time ##t##, would a distant observer perceive a warp bubble's interior volume as blown up, or would it seem compressed? Looking in the catalogue of spacetimes at the static local tetrad of the Alcubierre metric, the ##e^x_{(x)}## leads me to think that a static...
  36. Onyx

    A Proper Volume on Constant Hypersurface in Alcubierre Metric

    I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...
  37. mopit_011

    Estimating the Volume of a Cylindrical Shell

    Using the equation above, I plugged in 5.5 inches for the radiu and 0.5 inches for the value of dr and then solved for the estimate of the change in volume, dV. However, the solution instead uses a value of 6 inches for the radius receiving a different estimate for the problem than I did. Is my...
  38. B

    I Which units is this conversion factor for (molar volume)? 0.023901488

    I'm getting the wrong results when using an old, undocumented code and just realized there's a number lurking in it that I can't account for. It's: 0.023901488 and it is multiplied with molar volume and pressure. I have searched for a couple of hours but just can't figure out what the units...
  39. brotherbobby

    Spirit evaporating from a bowl

    Problem statement : I draw the problem statement above. I hope I am correct in inferring that the bowl is hemispherical. Attempt : I could not attempt to the solve the problem. We are given that the rate of change (decrease) in volume is proportional to the surface area ...
  40. yucheng

    I Do volume integrals involve bounding surfaces?

    In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents) \begin{align} \vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3}...
  41. LCSphysicist

    How Does Quantum Uncertainty Define the Position of a Positron?

    So, let's assumed ##v_i = (v_i \pm \Delta v_i)##. We can say that, for minimum values, $$\Delta X_i \Delta v_i = \hbar / (2m)$$. $$\Delta X_i = \frac{\hbar}{2 m \Delta v_i} \implies V_{min} = \Delta X \Delta Y \Delta Z = \frac{\hbar}{2 \Delta v_x m } \frac{\hbar}{2 \Delta v_y m} \frac{\hbar}{2...
  42. wruehl1

    Calculating fill rate if pressure delta, volume, and time are known?

    I am looking for an equation that I can use to compute L/min or mL/min for a 480cc vessel going from 150bar to 250bar with a fill time of 6min. Sensors for flow rate at these pressures are hard to find, but I thought there might be a way to work this out with the parameters known. An equation...
  43. warhammer

    Seeking Guidance to Find Surface & Volume Bound Charges of a Half Cone

    This was a trivial question I had (which I posted here on the PF EM Forum: https://www.physicsforums.com/threads/bound-charges-polarisation-of-a-half-cone.1015308/). As I received no response on the above link I decided to post the same as a self formulated HW problem. Below I have attached an...
  44. S

    Volume of pyramid formed by center of 5 spheres inside a hemisphere

    Let the radius of the small sphere = r 3r = 1 → r = 1/3 ##x=\sqrt{4r^2-r^2}=r\sqrt{3}## Volume of pyramid: $$=\frac{1}{3} \times (2r\sqrt{3})^2 \times r$$ $$=\frac{4}{27}$$ So m + n = 31, but the answer is 29. I guess my mistake is assuming line AB is tangent to the top sphere. How to do...
  45. WMDhamnekar

    Finding the volume of the solid

    My attempt : ## \displaystyle\int_0^2 \displaystyle\int_0^{2-x} (2-2x -2y) dy dx = -\frac43 ## But it is wrong.
  46. Ahmed1029

    I Average electrostatic field over a spherical volume

    this formula in the picture is the average electrostatic field over a spherical volume of radius R. It is the same expression of the electrostatic field, at the (position) of the point charge, of a volume of charge of uniform density whole entire charge is equal to (negative)q. My question is...
  47. MevsEinstein

    What has a shape but no volume?

    I was reading a Chemistry book when I read about the three states of matter. Everyone knows what they are, but I didn't know the simplest way to describe each of the three until I read this book. It said that a solid has a shape and a volume, a liquid has no shape but has a volume, and a gas has...
  48. Tertius

    I Computing Volume in General Relativity: Use of Tensor & Friedmann Eqns

    When we compute the stress energy momentum tensor ## T_{\mu\nu} ##, it has units of energy density. If, therefore, we know the total energy ##E## of the system described by ## T_{\mu\nu} ##, can we compute the volume of the system from ## V = E/T_{00}##? If it holds, I would assume this would...