Volume Definition and 100 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. 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...
  2. 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 ...
  3. Tertius

    I Computing volume in general relativity

    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...
  4. Istiak

    Calculate the volume change with gas temperature for this piston in a cylinder

    Solution attempt : Option : I am sure that my work is wrong. But, I must add solution attempt in PF that's why I just added that. How can I solve the problem?
  5. bobbyg

    Relative volume of combusted hydrogen

    Hi, If I had a volume of Brown's gas at 20°C / 1atm, what would the expected volume [of the resultant steam] be immediately after it was ignited? Thanks! Bob
  6. Adesh

    I Why volume is conserved but not the surface area?

    A water drop of radius ##10^{-2}## m is broken into 1000 equal droplets. Calculate the gain in surface energy. Surface Tension of water is ##0.075 ~N/m##. So, for the solution of the above problem we need to know how much surface area (combining all 1000 droplets) have increased from the...
  7. G

    Determing the center of gravity of a shaded section

    Determine the volume of the shaded area around the Y-axis by using the theorem of Pappus Guldinus, where value of R = 143,3 cm. a) Determine the area of the shaded section. b) Determine the center of gravity of the shaded section. c) Detrmine the volume by using the theorem of Pappus Guldinus...
  8. brotherbobby

    Weight of water with objects floating

    We understand that the crucial thing about the problem is that the volume of water present in the three containers are not the same. Also, we note that in each case the weight of the container is the total weight of its contents. (A student might be confused as to why should be so - after all...
  9. L

    Superionic and metallic states of water and the Anomaly of Water

    https://www.wired.com/story/a-bizarre-form-of-water-may-exist-all-over-the-universe/ Black ice I knew the Black Ice Theories since around 1990 https://www.nature.com/articles/s41586-019-1114-6 -- Demontis, P., LeSar, R. & Klein, M. L. New high-pressure phases of ice. Phys. Rev. Lett. 60...
  10. Aletag

    Work done BY the gas in a cyclic thermodynamic process

    Since the assignment asks the work done by the gas, that should be equal to P1*(V2-V1) aka the area under the P1 line. Do I have to subtract the work done to the system or is this the solution already? If so, why do I need P2?
  11. Michael_0039

    Find the volume of the solid formed by the rotation around the y=0

    Hi, I find this... Please tell me your opinion on this. Thanks.
  12. L

    Thermodynamic Systems - Volume, Final Pressure and Final Temperature

    Mass = 0.12kg Initial temp = 500°c = 773K Initial pressure = 0.8 MPa = 800,000 Pa Final volume = 90L R = 287 Jkg^-1K^-1 1) Initial Volume V=mRT/P 0.12 x 287 x 773 / 800,000 = 26,662.12m^3 2) Final Pressure P2 = P1P2^1.2/V2^1.2 800,000 x 26,662.12^1.2 / 90 = 1,816,095,330 Pa = 1,816 MPa 3)...
  13. J

    I Basic adiabatic expansion Question

    I have a compressed pure gas at a specific temperature and volume. (T1, V1) It suddenly (adiabatically) expands until it's at ambient pressure and a specific temperature. (P2, T2). Given: T1, V1, T2, and P2, I want to find P1 and V2. There's a great example in wikipedia which is almost...
  14. QuarkDecay

    A Volume of a closed Universe

    The final result must be V=2π2α3 Hint says we must use the dV in the spherical system (dV=r2sin2θdrdθdφ) as well as the equation of the three-dimensional metric ds2= c2dt2 - a2[ dr2/(1-kr2) +r2(dθ2 +sin2θ dφ2) ] For a closed universe we know k=+1 and with dt=0 My problem is, I don't understand...
  15. Lardos

    A Ideas for determining the volume of a rotating object

    Hello everybody, I am currently working on an experiment investigating the formation of planets. I have a vacuum chamber in which dust particles form bigger agglomerates through accretion (sticking together). From the imagery I can see those agglomerates which are build up by smaller...
  16. patric44

    Boyle's law problem -- A beaker containing air is pushed upside-down into water....

    Homework Statement a Basin contains water , a beaker is put upside down to a depth of 3m inside it , if the volume of the beaker is 250 cm^3 . and its C.S area = 200 cm2 calculate the length of the water column which rises inside the baker , supposed that their is no air leakage from the...
  17. T

    Attempt at volume integration to compute the full field equation

    I'm trying to figure out this volume integral, a triple integral, of a 9-variable function. 3 Cartesian-dimension variables, and 6 primed and un-primed co-ordinates. After the volume integration, the un-primed co-ordinates will have been gotten rid of, leaving a field function in terms of...
  18. Q

    I Deriving the spherical volume element

    I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
  19. hnnhcmmngs

    Equation of state of a solid

    Homework Statement [/B] Find the equation of state of a solid that has an isobaric expansion coefficient dV/dT = 2cT - bp and an isothermal pressure-volume coefficient dV/dp = -bT (Assume the solid has a volume Vo at zero temperature and pressure. Enter a mathematical equation. Use any variable...
  20. I

    How long will it take for a gas cylinder to empty?

    The main aim of my question is to find time taken by gas filled tank while emptying (from full to zero) it into atmosphere? I require a set of equations which might be used in the calculation of such a situation. Known parameters are: 1. Pressure inside the gas tank. 2. Dimension and shape of...
  21. bornofflame

    [MultiVarCalc] Find the volume of the solid

    Homework Statement Find the volume of the solid between the cone ##z = \sqrt{x^2 + y^2}## and the paraboloid ##z = 12 - x^2 - y^2##. Homework Equations ##x^2 + y^2 = r^2## The Attempt at a Solution I drew a simple diagram to start off with to visualize the solid formed by the intersection of...
  22. N

    A Volume of a 3-torus

    I have recently read that physicists are exploring the possibility that the universe is a 3-torus. I have been trying to find the formula for the volume of a three-torus online but cannot find it. Can anyone tell me the formula?
  23. E

    Why is internal energy not a function of pressure?

    I'm reading a book on thermal physics and the author says this: "In general, the internal energy will be a function of temperature and volume, so that we can write U =U(T,V) " How do we know this intuitively and how do we know that internal energy is not a function of pressure as well?
  24. K

    Pitch and amplitude of sound wave Vs its volume

    Hi. If I know the pitch and amplitude of a sound wave, will I be able to calculate its volume. I can understand volume of devices vary betwerb brands and other categories. For the sake of discussion, let's assume volume to be a consistent unit or if db is the right unit, let's take that.
  25. nightingale123

    I N-th dimensional Riemann integral

    Hello I have a question regarding something we wrote in class today. Let ##A## be a bounded subset of ##R^n##, let ##f,g:A\to \mathbb{R}## be integrable functions on A. ##a)## if ## A## has a volume and ##\forall x \in A :m\leq f(x) \leq M## then ##mV(A)\leq \int_{A}f(x)\leq MV(A)## this...
  26. M

    Proving the equation for the height of a cylinder

    Homework Statement Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed. Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the...
  27. K

    Chain rule in thermodynamics

    I'm trying to understand why $$\left(\frac{\partial P}{\partial T}\right)_V = -\left(\frac{\partial P}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_P$$ where does the minus sign come from?
  28. M

    B An infinitly dense human body

    Hello, I've been watching too much anime. I know that isn't the best way to ask a question scientific in nature, but visualizing bizarre situations is one of the best ways to come up with interesting questions. Anyways, in order to break a rock by compressing it with your hands, you would need...
  29. Q

    I Shear Rate in a Rectangular Channel

    Say you have a rectangular channel with a width, w (m), and a height, h (m) and an infinite length. The channel itself is fixed and none of the sides can move with respect to one another. If you know the volume flow rate, V (m3/s) of fluid through the channel, how do you calculate the shear rate...
  30. M

    B Definition of Volume

    Suppose if we have a cube: The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see...
  31. HelloCthulhu

    What is the molality of 1M NH4OH in 70.092ml of water?

    Homework Statement What is the molality of 1M NH4OH dissolved in 70.092ml of water? What is the total mass of the solution? What is the total volume? What is the concentration percent? What is the molar concentration? 1M NH4OH = 35.046g density of water = 1g/ml density of NH4OH = 0.88 g/ml...
  32. psuedoben

    Measure the volume of a lighter than air balloon

    hello, i am trying to calculate the volume of a balloon (which is quite large). It has been filled with helium via a valve connecting helium storage tanks to the balloon. The knowns I have are the volume of the storage tanks, the intital pressure in the tanks, and the final pressure in the tanks...
  33. L

    Volume of a Solid using Cylindrical Shells

    Homework Statement Find the volume of the region bounded by the curves y=3x-2, y=6-x, and the x-axis when the region is rotated around the y-axis. Homework Equations Volume using cylindrical shells: 2π∫r(x)h(x)dx The Attempt at a Solution I graphed the curves and then found the x-intercept...
  34. I

    B How can Greenland lose 269b tons/year and loose only 269km3?

    The density of ice is 920 kg/m3. Now let's calculate. m = 269 000 000 000 000 kg p = 920 kg/m3 ----------------------------- v=? V=m/p V=269 000 000 000 000 kg / 920 kg/m3 V = 292 391 304 347 m3 = 292 391 304.347 km3 v > 269 km3 *NOTE* I made a mistake in the title. I meant to write 269 km3...
  35. J

    Concentration of a salt produced by neutralization reaction

    Homework Statement 10.00 cm3 of 1.00 mol dm–3 sulfuric acid is fully neutralized by 20.00 cm3 of 1.00 mol dm–3 of sodium hydroxide. What is the concentration, in mol dm–3, of sodium sulfate solution produced by the reaction? A 0.33 B 0.50 C 0.67 D 1.00 Correct answer = A Homework Equations...
  36. U

    B Integrating to find surface area/volume of hemisphere

    To find the surface area of a hemisphere of radius ##R##, we can do so by summing up rings of height ##Rd\theta## (arc length) and radius ##r=Rcos(\theta)##. So the surface area is then ##S=\int_0^{\frac{\pi}{2}}2\pi (Rcos(\theta))Rd\theta=2\pi R^2\int_0^{\frac{\pi}{2}}cos(\theta)d\theta=2\pi...
  37. wasup23

    I Introducing two identical containers with different gasses

    If you have two different containers filled with two different gasses at the same temperature, would they have less pressure when connected to each other? Dalton's law states that each of the gases behave independently when it comes to pressure as they fill the space as if they were the only gas...
  38. K

    Volume of water required to cool thermal/nuclear plants?

    Homework Statement In the year 2004 the USA produced 1787 TWh of electrical energy in conventional thermal plants and 476 TWh in nuclear plants. Assuming 30% efficiency for nuclear plants and 40% for conventional thermal plants, determine the (annual) volume of cooling water required to cool...
  39. G Cooke

    I Humidity in a closed system

    We know that for a closed system, isobaric heating decreases the humidity and isothermal compression increases the humidity. But assuming that we start with the volume completely filled with humid air, is it not true that any increase in temperature must also increase the pressure? If so...
  40. P

    I Should we consider negative axis when finding the volume?

    Question: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y2 = x; x = 2y; about the y-axis Solution...
  41. C

    Finding Pressure change as height changes with Pfinal given

    Hi, I am doing a project for school where I am calculating at what height a balloon would pop from pressure while rising. At the point I am now, I have the initial pressure and the pressure it would pop at. I know work = pressure * change in volume, and that at some point I will need to use the...
  42. O

    Double integral, find volume of solid

    Homework Statement Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders: y = 1 − x2, y = x2 − 1 and the planes: x + y + z = 2 4x + 5y − z + 20 = 0 Homework Equations ∫∫f(x,y) dA The Attempt at a Solution So I solved for z in the plane...
  43. E

    How much power required to generate a particular pressure

    Are there any general guidelines or rule of thumb regarding how much power in watts would be required to pressurize a cylinder to a particular pressure and can I use this as a general rule, For eg. I want to pressurize a vessel of volume 1 cu. m how much power would be required and can we...
  44. E

    Collecting gas over water with a eudiometer

    Homework Statement When, using a eudiometer in a lab to collect gas over water, if you have to switch eudiometers is the final volume of water displaced equal to the water displaced in the first tube + the water displaced in the second, and would the pressure be calculated through the total...
  45. G

    Compressible air flow from a tank through a regulator

    This is probably a simple question but I am really struggling as to where to start. We have a tank of compressed air, Volume of tank (V1) = 0.2litres Pressure of tank (P1) = 206.8 bar Regulator Pressure (P2) = 58.6 bar I would like to find the volume of air that can pass through the regulator...
  46. Oannes

    Finding Distance given accleration and Standard Gravity.

    Homework Statement A car has a performance of 10,705 miles per gallon. If the car is given 100 grams of ethanol (Standard Gravity = 0.789) and drive until the fuel runs out, how far with they go in meters? Homework Equations Standard Gravity = Density of the Object /Density of Water Density =...
  47. kamhogo

    Syringe: Find force given volume and pressure

    Homework Statement A 60 mL syringe filled with air is connected to a pressure sensor. The latter reads 100kPa. You then push on the piston until it stops moving. You now have 30 mL and the pressure sensor reads 200kPa. What force is used to keep the syringe rubber at the 30 mL mark...
  48. G

    Propagation of Uncertainty - Hollow Cylinder

    Hi, I'm having quite a bit of trouble finding the propagation of uncertainty (using partial derivatives) of the volume of a hollow cylinder. The examples in my tutorial only demonstrates how to find the propagation of uncertainty on simple operations such as x + y, x/y, etc... 1. Homework...
  49. Oannes

    Finding Surface Area in square feet with Volume & Thickness

    Homework Statement How large a surface area in units of square feet will 1 gallon of paint cover if we apply a coat of paint that is 0.1cm thick? Homework Equations Since Volume is L * W * H and we can assume the object is square besides the height which in this case will be the thickness. So...
  50. E

    Enthelpy problems....

    In a trasformation in which P=costant, but internal pressure is different from external pressure, ΔH=Q? I'm asking this question because I know that Q=ΔU+PΔV (where P is the external pressure) and H=U+PV (where P is the sistem pression, so the internal pressure) Am I right?